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<!DOCTYPE archimedes [ <!-- footnotes and margnotes get yanked out of their <p>s, repl. by <arrow>, --> <!-- and moved to own p. --> <!ELEMENT foot.target (#PCDATA) > <!-- e.g. number in front of a footnote --> <!ATTLIST foot.target id ID #REQUIRED n CDATA #IMPLIED > <!ELEMENT margin.target (#PCDATA) > <!ATTLIST margin.target id ID #REQUIRED n CDATA #IMPLIED > <!ELEMENT arrow.to.target (#PCDATA)* > <!ATTLIST arrow.to.target symbol CDATA #IMPLIED id ID #IMPLIED n CDATA #IMPLIED> <!-- reference to removed figure/note. contains mark in text. --> <!-- n attribute contains id of referent target --> <!-- Can marginalia ever have referring marks in text? --> <!ELEMENT archimedes (info, text) > <!ATTLIST archimedes id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT author (#PCDATA) > <!ATTLIST author id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT back ( section | pb?)+ > <!ATTLIST back id ID #IMPLIED n CDATA #IMPLIED type CDATA #IMPLIED> <!ELEMENT body (chap, pb?)+ > <!ATTLIST body id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT chap ( p | pb | figure | table)+ > <!ATTLIST chap id ID #IMPLIED n CDATA #IMPLIED type CDATA #IMPLIED> <!ELEMENT chunk (#PCDATA) > <!ATTLIST chunk id ID #IMPLIED n CDATA #IMPLIED unit CDATA #IMPLIED > <!ELEMENT date (#PCDATA) > <!ATTLIST date id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT editor (#PCDATA) > <!ATTLIST editor id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT emph EMPTY> <!ATTLIST emph type (italics|bold|sup|sub|over|smallcaps|center|roman|ul|quote|other) "italics" id ID #IMPLIED n CDATA #IMPLIED> <!ELEMENT emph.end EMPTY> <!-- the type attrib. here shld be entitized --> <!ATTLIST emph.end type (italics|bold|sup|sub|over|smallcaps|center|roman|ul|quote|other) "italics" id ID #IMPLIED n CDATA #IMPLIED> <!ELEMENT expan (#PCDATA | foot.target|margin.target|arrow.to.target|pb|lb|emph|emph.end|gap)* > <!ATTLIST expan id ID #IMPLIED n CDATA #IMPLIED abbr CDATA #IMPLIED type CDATA #IMPLIED > <!ELEMENT figure (#PCDATA) > <!ATTLIST figure id ID #IMPLIED place (margin|text) "text" xmlns:xlink CDATA #FIXED "http://www.w3.org/1999/xlink" xlink:type (simple) #FIXED "simple" xlink:href CDATA #IMPLIED> <!ELEMENT foreign (#PCDATA | expan | foot.target|margin.target|arrow.to.target|pb|lb|emph|emph.end|gap)* > <!ATTLIST foreign id ID #IMPLIED n CDATA #IMPLIED lang CDATA #IMPLIED> <!-- <foreign> for text within sentences not in the main lang of the text --> <!-- or for text within non-default-lang higher elements --> <!ELEMENT front ( section | pb?)+ > <!ATTLIST front id ID #IMPLIED n CDATA #IMPLIED type CDATA #IMPLIED> <!ELEMENT gap EMPTY > <!ATTLIST gap id ID #IMPLIED n CDATA #IMPLIED desc CDATA #IMPLIED> <!-- gap is a catch-all tag employed at Perseus and in the TEI to isolate --> <!-- uncertain DE markup. It appears here for continuity's sake. --> <!ELEMENT info (author, title, date, place, editor, publisher, translator, lang, chunk, locator?) > <!-- how many of these should be required? --> <!-- what about bringing in line with dublin core? --> <!ATTLIST info id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT lang (#PCDATA) > <!ATTLIST lang id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT locator (#PCDATA) > <!ATTLIST locator id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT lb EMPTY > <!ATTLIST lb ed CDATA #IMPLIED id ID #IMPLIED n CDATA #IMPLIED> <!-- <lb> occurs at <s> level and at <p> level --> <!-- <lb> at end of <s> must be placed after </s> --> <!-- unrecognized symbols will appear inline acc. to special conventions. --> <!-- (as in DE specs) --> <!-- emph and and emph.end are elements that are quer to the xml structure --> <!ELEMENT p (s |pb|lb|emph|emph.end|gap )+ > <!ATTLIST p id ID #IMPLIED n CDATA #IMPLIED type (main|marked|foot|margin|list|head|caption|table) "main" > <!-- captions go in paragr's immediately after figure, with arrow.to.target --> <!-- pb can occur within p but NOT at the start or end --> <!ELEMENT pb EMPTY > <!ATTLIST pb ed CDATA #IMPLIED id ID #IMPLIED n CDATA #IMPLIED pagenum CDATA #IMPLIED> <!-- the "ed" attribute gives the edition in which the page break occurs. If no --> <!-- edition is given, it is assumed to be that given in info --> <!-- pagenum gives the page number actually printed on the page. --> <!-- n gives the page number numbered consecutively from page 1 of text. --> <!-- pb can occur inside <s> etc. and inside <p> and inside <chap> --> <!-- <pb> at end of <p> must be placed after </p> --> <!ELEMENT place (#PCDATA) > <!ATTLIST place id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT publisher (#PCDATA)* > <!ATTLIST publisher id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT s (#PCDATA| foreign | expan | foot.target|margin.target|arrow.to.target|pb|lb|emph|emph.end|gap)* > <!ATTLIST s id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT section ( p | pb | figure)+ > <!ATTLIST section id ID #IMPLIED n CDATA #IMPLIED type CDATA #IMPLIED> <!ELEMENT table ( row+ ) > <!ATTLIST table id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT row ( cell+ ) > <!ATTLIST row id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT cell (#PCDATA| foreign | expan | foot.target|margin.target|arrow.to.target|pb|lb|emph|emph.end|gap)* > <!ATTLIST cell id ID #IMPLIED n CDATA #IMPLIED > <!ELEMENT text (pb?, front, pb?, body, pb?, back) > <!-- if front and back are going to be optional, then pb can be allowed to --> <!-- occur only within front body back. --> <!ATTLIST text type CDATA #IMPLIED id ID #IMPLIED n CDATA #IMPLIED> <!ELEMENT title (#PCDATA) > <!ATTLIST title id ID #IMPLIED n CDATA #IMPLIED type CDATA #IMPLIED > <!ELEMENT translator (#PCDATA)* > <!ATTLIST translator id ID #IMPLIED n CDATA #IMPLIED > <!ENTITY shy "[-]"> <!ENTITY aacute "[aacute]"> <!ENTITY aelig "[aelig]"> <!ENTITY agrave "[agrave]"> <!ENTITY amacr "[amacr]"> <!ENTITY amp "[amp]"> <!ENTITY atilde "[atilde]"> <!ENTITY auml "[auml]"> <!ENTITY eacute "[eacute]"> <!ENTITY egrave "[egrave]"> <!ENTITY emacr "[emacr]"> <!ENTITY etilde "[etilde]"> <!ENTITY euml "[euml]"> <!ENTITY gt "[gt]"> <!ENTITY iacute "[iacute]"> <!ENTITY igrave "[igrave]"> <!ENTITY imacr "[imacr]"> <!ENTITY lbrk "[lbrk]"> <!ENTITY ldquo "[ldquo]"> <!ENTITY longs "[longs]"> <!ENTITY lt "[lt]"> <!ENTITY ograve "[ograve]"> <!ENTITY otilde "[otilde]"> <!ENTITY rdquo "[rdquo]"> <!ENTITY sacute "[sacute]"> <!ENTITY shy "[shy]"> <!ENTITY szlig "[szlig]"> <!ENTITY ugrave "[ugrave]"> <!ENTITY umacr "[umacr]"> <!ENTITY utilde "[utilde]"> ]> <archimedes> <info> <author>Guidobaldo del Monte</author> <title>Le Mechaniche</title> <date>1581</date> <place>Venezia</place> <editor></editor> <publisher></publisher> <translator></translator> <lang>it</lang> <chunk unit="page*">page</chunk> </info> <text> <front> </front> <body> <chap> <pb id="p.0001"/> <p type="head"> <s>LE <lb/>MECHANICHE <lb/>DELL'ILLVSTRISS SIG.</s> </p> <p type="head"> <s>GVIDO VBALDO <lb/>DE' MARCHESI DEL <lb/>MONTE: <lb/>TRADOTTE IN VOLGARE <lb/>DAL SIG. FILIPPO PIGAFETTA: <lb/>Nellequali &longs;i contiene la vera Dottrina di tutti gli I&longs;trumenti <lb/>principali da mouer pe&longs;i grandis&longs;imi con <lb/>picciola forza.</s> </p> <p type="head"> <s><emph type="italics"/>A beneficio di chi &longs;i diletta di que&longs;ta nobili&longs;&longs;ima scienza; & ma&longs;&longs;imamente <lb/>di Capitani di guerra, Ingegnieri, Architetti, & d'ogni <lb/>Artefice, che intenda per via di Machine <lb/>far opre marauiglio&longs;e, e qua&longs;i <lb/>&longs;opra naturali.<emph.end type="italics"/></s> </p> <p type="head"> <s>Et &longs;i dichiarano i vocaboli, & luoghi più difficili.</s> </p> <figure></figure> <p type="head"> <s><emph type="italics"/>In Venetia, Appre&longs;&longs;o France&longs;co di France&longs;chi Sane&longs;e. </s> <s>MD LXXXI.<emph.end type="italics"/></s> </p> <pb/> <figure></figure> <p type="head"> <s>ALL'ILLVSTRISSIMO <lb/>SIGNOR GIVLIO <lb/>SAVORGNANO, <lb/>CONTE DI BELGRADO. &c.</s> </p> <p type="head"> <s>Signore o&longs;&longs;eruandi&longs;&longs;imo.</s> </p> <figure></figure> <p type="main"> <s>C<emph type="italics"/>onciosia co&longs;a, che la &longs;cienza delle Mecha­<lb/>niche gioui &longs;ommamente à molte, & importan­<lb/>ti attioni della no&longs;tra vita, à gran ragione fu ella <lb/>da i Filo&longs;ofi, & da i Rè antichi &longs;timata degna di <lb/>laudi &longs;ingularißime; & i Matematici vi han­<lb/>no impiegato lo &longs;tudio, & l'opera più che meza­<lb/>namente, & i Principi fauoriti gl'ingegnieri ec <lb/>cellenti, & arricchiti. </s> <s>Ben è per certo di alti&longs;­<lb/>&longs;ima &longs;peculatione, & di &longs;ottile manifattura; imperoche tocca quella par­<lb/>te della Filo&longs;ofia, che tratta de gli elementi in vniuer&longs;ale, & del moto, & <lb/>della quiete de' corpi, &longs;econdo i luoghi &longs;uoi, a&longs;&longs;egnando la cagione in certo <lb/>modo de' loro mouimenti naturali; & anco sforzandoli, per via di machi­<lb/>ne à partir&longs;i da proprij &longs;iti, gli tra&longs;porta all'insù, & per ogni lato in mo­<lb/>uimenti contrari alla natura loro.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Mena ella ad effetto ambedue que&longs;te intentioni con le propo&longs;itioni che <lb/>na&longs;cono, & &longs;ono congiunte con la materia &longs;te&longs;&longs;a, & co' difici, & i&longs;trumen <lb/>ti, che forma artificialmente. </s> <s>La onde egli è dibi&longs;ogno con&longs;iderare que&longs;ta<emph.end type="italics"/> <pb/><emph type="italics"/>dottrina in due manìere; l'vna ìn quanto và &longs;peculando, & con ragione <lb/>di&longs;correndo &longs;opra le co&longs;e, che s'hanno à fare, &longs;eruendo&longs;i dell' Arithmetica, <lb/>della Geometria, dell' A&longs;trologia, & della Filo&longs;ofia naturale: & l'altra che <lb/>po&longs;cia le manda ad e&longs;ecutione, & haue nece&longs;sita dell'e&longs;&longs;ercitio, & lauoro <lb/>delle mani, v&longs;ando l'Architettura, la Pittura, il di&longs;egno, l'arte de' fabri, <lb/>de'legnaiuoli, de'muratori, & d'altri me&longs;tieri tali, per modo che ella vie­<lb/>ne ad e&longs;&longs;ere me&longs;colata, & in parte compo&longs;ta della naturale Filo&longs;ofia, delle <lb/>Matematiche, & delle arti manuali. </s> <s>Per laqual co&longs;a chiunque &longs;i troua <lb/>dotato d'ingegno acuto, & da fanciullo hà incominciato ad apprendere le <lb/>già dette &longs;cienze, & &longs;a di&longs;egnare, & lauorare di &longs;ua mano, potrà nel vero <lb/>ottimo Mechanico, & <expan abbr="inuētore">inuentore</expan>, & facitore di opere marauiglio&longs;e riu&longs;cire.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Infinite parti, & vtilißime à gli huomini comprende que&longs;ta noti­<lb/>tia, & in guerra, & in pace, ne i commodi della città, della villa, & della <lb/>mercatantia, & in altri; peroche la Medicina toglie da lei i difici per ri­<lb/>porre le o&longs;&longs;a &longs;mo&longs;&longs;e, & rotte ne i &longs;iti &longs;uoi. </s> <s>Onde pone Oriba&longs;io nel libro delle <lb/>Machine, diuer&longs;i i&longs;trumenti pre&longs;i dalla Mechanica, & <expan abbr="cõuertiti">conuertiti</expan> nell'v&longs;o del <lb/>la Medicina, come il Tri&longs;pa&longs;ton di Archimede: l'arte del nauigare ricono­<lb/>&longs;ce anco diuer&longs;i aiuti, come il timone, co'l quale, collocato di dietro, ouero <lb/>alle bande del nauilio ageuolmente lo moue, & dirizza, quantunque per <lb/>ri&longs;petto à tutto il corpo del va&longs;ello piccioli&longs;simo &longs;ia. </s> <s>I remi, che à gui&longs;a di <lb/>leua lo &longs;pingono innanzi, & l'arbore, & la vela &longs;ono pur di &longs;ua inuentio <lb/>ne. </s> <s>I molini, i quali &longs;i girano co'l vento, con l'acqua, & con la forza vi­<lb/>ua: & i pi&longs;trini, le carra, gli aratri, & altri ordigni di villa; il pe&longs;are con <lb/>la bilancia, & con la &longs;tadera; il cauare l'acqua da pozzi con le grù, ouero <lb/>cicogne, dette da latini to&longs;senoni, che &longs;ono come grandis&longs;ime bilancie, & <lb/>con le rote, & altre co&longs;e tali &longs;i riducono alla Mechanica. </s> <s>La ragione pa­<lb/>rimente del condurre le acque, & da profondis&longs;ime valli in alto farle &longs;ur <lb/>gere uà &longs;otto lei. </s> <s>Chiamarono gli antichi coloro Mechanici ancora, i quali <lb/>co'l fiato, ò vento, ouero acqua, ò corde, ò nerui faceuano vedere, & vdi <lb/>re effetti miracolo&longs;i; come &longs;uoni diuer&longs;i, & canti d'augelli, & fin ad e&longs;pri­<lb/>mere la voce humana in parole: & quelli che con horologi, i quali &longs;i mo­<lb/>uono da &longs;e &longs;tes&longs;i con rote, ò da acqua, ò da &longs;ole il tempo mi&longs;urarono, & di­<lb/>&longs;tin&longs;ero in hore. </s> <s>Appartengono alla Mechanica gli facitori delle Sfere <lb/>compartite ne'&longs;uoi cieli, co'l mouimento de'Pianeti, & di tutti i corpi <lb/>cele&longs;tiali à &longs;embianza dell'vniuer&longs;o mondo, & ciò mediante il mouimen­<lb/>to eguale, & in giro, che loro daua l'acqua, di cui la fama &longs;uona e&longs;&longs;ere <lb/>&longs;tato Archimede Siracu&longs;ano il primo mae&longs;tro. </s> <s>il mouere etiandio con poca<emph.end type="italics"/> <pb/><emph type="italics"/>forza pe&longs;i grandis&longs;imi con i&longs;trumenti, & ingegni diuer&longs;i è principale of­<lb/>ficio della Mechanica, come Bilancie, Stadere, Leue, Taglie, Cunei, Moli­<lb/>nelli, Rote co' denti & &longs;enza, Viti d'ogni &longs;orte, Argani, Mangani, Triuel <lb/>le, & altri molti, i quali da que&longs;ti &longs;i compongono: & &longs;econdo Ari&longs;totele <lb/>tutti &longs;iriducono alla Leua, & al cerchio, & alla machina ritonda, laquale <lb/>quanto è maggiore, tanto più velocemente &longs;i moue. </s> <s>L'arte del fortificare <lb/>le piazze, & i &longs;iti, & del difendergli, laquale acconciamente &longs;i puote chia <lb/>mare Architettur a militare, è pro&longs;es&longs;ione Mechanica: peroche per via di <lb/>Cortine, & di Baloardi, & d'altri ripari, qua&longs;i con ma<gap/>hine, & i&longs;tr umen­<lb/>ti s'ingegna l'huomo con po hi &longs;oldati di ributtarne in dietro molti, & <lb/>mantener&longs;i con vantaggio. </s> <s>Il fabricare, & adoprare oltre à c<gap/>ò gli i&longs;tru <lb/>menti da guerra è proprio dono di que&longs;ta &longs;cienza, come Bali&longs;te, ò Bale&longs;tre, <lb/>Catapulte, Scorpioni, Fionde, & &longs;imili, che da lontano gittano foco, & &longs;aßi, <lb/>& ma&longs;&longs;e di ferro pe&longs;anti dugento cinquanta, & più libre, & Moli da <lb/>molino &longs;econdo Silio Italico, & Vitruuio, per di&longs;tanza di for&longs;e<emph.end type="italics"/> 300. <emph type="italics"/>pas&longs;i <lb/>à mi&longs;ura con ruin &longs;o colpo; & &longs;aette, & verettoni, & falariche grandi <lb/>à gui&longs;a di traui: & quelli che percoteudno con l'vrto da pre&longs;&longs;o, come Arie <lb/>ti, Onagri, Te&longs;tugini, & &longs;imili; & in altri v&longs;i, come <expan abbr="Sãbuche">Sambuche</expan>, Corui, Mani <lb/>di ferro, & gli altri maritimi, & Angoni, Monangoni, Tollenoni, &longs;cale &longs;no­<lb/>date, ponti, torri mobili, & &longs;imili difici antichi, i quali &longs;ono &longs;tati poi ri <lb/>fiutati, &longs;uccedendo in &longs;uo luogo le Artiglierie, da e&longs;&longs;ere anch'e&longs;&longs;e ordinate <lb/>nell' ampiezza della con&longs;ideratione Mechanica, facendo elle còn sì poca mæ <lb/>teria acce&longs;a, tanto horribile perco&longs;&longs;a.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Que&longs;ta &longs;cienza, che fuor di quanto &longs;i è detto, abbraccia innumerab ili <lb/>altri v&longs;i, & diletteuoli, & nece&longs;&longs;ari à mortali, in diuer&longs;i tempi hebbe in <lb/>&longs;orte vari &longs;tati, per ri&longs;petto à gli artefici, che la e&longs;ercitarono: peroche, <lb/>di là cominciando, ne gli antichis&longs;imi &longs;ecoli, che pa&longs;&longs;arono auanti la guer <lb/>ra di Troia vi&longs;&longs;e Dedalo Athenie&longs;e gran mae&longs;tro di Mechanica, ilquale <lb/>trouò il primiero la &longs;ega, l'a&longs;cia, il piombino da torre le diritture, la tri­<lb/>uella, l'albero, l'antenna, la vela, & altri or digni: di&longs;egnò in Creta poi <lb/>quell'intricato labirinto, & alla fine gli conuenne fabricare per &longs;e, & per <lb/>Icaro &longs;uo figlio due paia d'ali, & volar&longs;ene via per l'aere à gui&longs;a d'au­<lb/>gelli, come cantano i Poeti.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Nella fabrica del tempio di Salomone, che fu la maggiore per grandez <lb/>za, per mae&longs;tria d' Architettura, & ornamento, di quanie ne &longs;iano &longs;tate <lb/>fatte giamai; & delle piramidi, & di tanti altri difici di quei &longs;eco'i, che <lb/>hanno riempito il mondo di &longs;tupore, egli &longs;i può credere, che interueni&longs;&longs;ero<emph.end type="italics"/> <pb/><emph type="italics"/>curamente: & del pre&longs;entare al nemico il fatto d'arme con vantaggio: <lb/>Del fortificare, & difendere i &longs;iti, & offenderli con le mine, con le trin­<lb/>cee, con le artiglierie, con gli a&longs;&longs;alts, & con tutti gli altri sforzi; & d'o­<lb/>gni parte della militare &longs;cienza.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ritornati in pace i Prencipi Chri&longs;tiani, &longs;i dedicò al &longs;eruigio de' Sereniß. <lb/>&longs;uoi Signori, oue ne i più importanti carichi, & maggiori, & in due guer <lb/>re haue e&longs;&longs;a aggiunto cinquanta anni di noua, & ottima &longs;eruitù all'an­<lb/>tica di qua&longs;i dugento anni, continua, & fedeli&longs;s. </s> <s>fattagli da i &longs;uoi pre­<lb/>dece&longs;&longs;ori Sauorgnani, fabricando nello &longs;patio di que&longs;to tempo in diuc<gap/>&longs;e pro <lb/>uincie de' &longs;uoi &longs;tati pre&longs;&longs;o che cinquanta Baloardi, con eccellentißima ra­<lb/>gione inte&longs;i, & con vero magi&longs;terio lauorati, & notabilißimo ri&longs;parmio <lb/>del publico danaro.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma per tornare alle Mechaniche dico, che quando gli anni pa&longs;&longs;ati io <lb/>venni à vi&longs;it arla ad O&longs;opo &longs;ua fortezza, &longs;entì &longs;ommo piacere in &longs;corgere <lb/>quel monte, che circonda più d'un miglio, &longs;ituato alla foce del fiume Ta­<lb/>gliamento, oue dalle &longs;trettezze di quei gioghi s'allarga nelle pianure del <lb/>Friuli, d'ogn'intorno alto pre&longs;&longs;o che &longs;e&longs;&longs;anta paßi à mi&longs;ura, tutto di ma­<lb/>cigno duro, & di&longs;co&longs;ce&longs;e, & erto sì, che rende la &longs;alita impoßibile fornito <lb/>attorno di baloardi cauati nel &longs;a&longs;&longs;o, & di molti tagli, & canoniere per <lb/>ferire gli aduer&longs;ari, & di artiglierie, & d'arme d'ogni &longs;orte à &longs;u&longs;&longs;icienza, <lb/>da cui &longs;i hà vi&longs;ta di qua&longs;i tutto il Friuli, & è &longs;<gap/>udo, & riparo, come al­<lb/>tra volta fù, contra l'empito delle genti nemiche, lequali in Italia tenta&longs; <lb/>&longs;ero di &longs;oendere da quella parte; po&longs;to di co&longs;ta alla &longs;trada principale, che <lb/>conduce in Lamagna, per laqual vanno, & vengono Signori, & Principi, <lb/>& Amba&longs;ciadori, & infinite mercatantie; onde ella, che tiene &longs;empre le <lb/>guardie, & vedette sù quel monte, quando pa&longs;&longs;ano Signori principali, <lb/>hà per co&longs;tume di &longs;alutargli con le &longs;ue artiglterie, & conuitarg'i anco <lb/><gap/>el &longs;uo alloggiamento d'O&longs;opo, oue tutto l'anno &longs;oggiorna, quantunque <lb/>habbia & Belgrado, & Aris, & Ca&longs;telnouo, & Sauorgnano, & villaggi <lb/>a&longs;&longs;ai: percioche l'aere vi è purißimo, & &longs;pende il &longs;uo tempo in ocio con ne <lb/>gocio, di continuo vi&longs;itata da Gentil'huomini, & Signori diuer&longs;i; ta che la <lb/>&longs;ua ca&longs;a viene ad e&longs;&longs;ere vn ridotto di per&longs;one virtuo&longs;e, & vn'albergo di &longs;ol <lb/>dati, & di dottori. </s> <s>lui &longs;i caualca, tenendo ella vna &longs;talla piena di buoni&longs;­<lb/>&longs;imi caualli, &longs;i armeggia, &longs;i và alla caccia, & in ogni attione &longs;i e&longs;ercita vi­<lb/>ta cauallere&longs;ca. </s> <s>Oltre à quanto hò diui&longs;ato, pre&longs;i anco diletto in vedere la <lb/>&longs;ua habitatione e&longs;&longs;ere à gui&longs;a d'vna bottega d'arme politamente à &longs;uoi <lb/>luoghi &longs;erbate: & vn magazino di machine bellico&longs;e, & da mouer pe&longs;i,<emph.end type="italics"/> <pb/><emph type="italics"/>eccellenti Mechanici, per leuare in alto le pietre &longs;mi&longs;urate, & per altre <lb/>opere, lequali à condurgli à fine &longs;i ricercauano. </s> <s>Nacquero dapoi Eudo&longs; <lb/>&longs;o, & Archita Tarentino, ambidue valenti ingegnieri; & di Archita &longs;i <lb/>legge, che lauorò di legno vna colomba con tanta mae&longs;tria temperata, & <lb/>gonfiata, che da &longs;e volaua per l'aria à gui&longs;a di viua colomba. </s> <s>Seguì co­<lb/>&longs;toro il Filo&longs;ofo Ari&longs;totele, ilquale certe poche, ma bellis&longs;ime que&longs;tioni Me­<lb/>chaniche, la&longs;ciò &longs;critte. </s> <s>A lui venne appre&longs;&longs;o Demetrio Rè, nominato il <lb/>pigliatore, ò di&longs;truggitore delle città, peroche fabricaua machine, & difi­<lb/>ci, co' quali per di&longs;opra vi montaua, & &longs;e ne faceua padrone, lequali per <lb/>auentura furono &longs;imiglianti alla machina detta Cauallo, con cui li Greci <lb/>pre&longs;ero la famo&longs;a Troia; di che ragionando Pau&longs;ania nell' Attica, dice che <lb/>giudica e&longs;pre&longs;&longs;a mattezza il credere, che fo&longs;&longs;e vn cauallo, & non machina <lb/>bellico&longs;a per acco&longs;tare alle muraglie, & prenderle. </s> <s>Que&longs;to Rè cominciò <lb/>ad aumentare la Mechanica in qualche honore. </s> <s>Ma Archimede, che fù <lb/>il megliore artefice di quanti fecero giamai que&longs;ta profes&longs;ione innanzi, & <lb/>dopo lui, & qua&longs;i vn lume, che poi ha illu&longs;trato tutto il mondo, accrebbe <lb/>in colmo la riputatione della Mechanica, & di pouera arte, & vile, che pri <lb/>ma era, come vuole Plutarco nella vita di Marcello, nel numero delle arti <lb/>nobili, & pregiate alla militia pertinenti la ripo&longs;e. </s> <s>Imperoche combat­<lb/>tendo Marcello Sir acu&longs;a patria &longs;ua per mare, & per terra con grande <lb/>ho&longs;te di Romani, egli co'&longs;uoi diuer&longs;i ingegni, & machine differenti, ribut­<lb/>tò &longs;empre gli sforzi, con graue lor danno, & vergogna; come Liuio, Plutar <lb/>co, & altri nominando i difici che v&longs;aua, diffu&longs;amente raccontano. </s> <s>Per­<lb/>cioche quando Marcello s'auicinaua aile muraglie per conqui&longs;tar le con la <lb/>Sambuca, il buon Archimede co'l Tollenone, & con le mani di ferro la al­<lb/>zaua di pe&longs;o in aere, & poi &longs;nodando quegli vncini &longs;uoi, la faceua cadere <lb/>da alto, in mare &longs;ommergendola; il mede&longs;imo effetto adoprando eontra gli <lb/>altri nauili, sì fattamente, che gli conuenne allontanare l'armata ben to <lb/>&longs;to dalle mura. </s> <s>Ne ce&longs;sò tuttauia d'infe&longs;tare il nemico: ma &longs;i come nota <lb/>Galeno nel terzo libro de' temper amenti, & Giouanni Zonara, & Tze&longs;es <lb/>confermano, allegando Diodoro, & Dione, compo&longs;e certi &longs;pecchi grandi <lb/>& concaui, &longs;econdo la proportione della di&longs;t anza di quei va&longs;elli dalla mu­<lb/>raglia, & opponendogli à raggi del Sole in diritta linea qua&longs;i per miraco­<lb/>lo, gli bru&longs;ciaua. </s> <s>Dalla parte della terra &longs;imilmente offendeua gli aduer­<lb/>&longs;ari con arme diuer&longs;e da gittare. </s> <s>Per laqual co&longs;a nè in mare, nè in terra <lb/>da gl'ingegni di quell'eccellente Mechanico &longs;i poteua egli &longs;chermire, nuoui <lb/>ripari, & horribili offe&longs;e apparecchiando &longs;empre. </s> <s>Pappo Ale&longs;&longs;andrino<emph.end type="italics"/> <pb/><emph type="italics"/>allega il quar ante&longs;imo trouato <gap/> Archimode, per dichiarare, che almeno <lb/>i &longs;uoi difici al numero di quaranta a&longs;cendeuano. </s> <s>La onde Marcello, veg­<lb/>gendo, che niuno profitto apportauano all'impre&longs;a gli a&longs;&longs;alti &longs;uoi, & che <lb/>erano vn mettere le genti ad euidente pericolo, per cagione di quel &longs;olo <lb/>valoro&longs;o vecchio, gli nacque vna tal opinione, & à tutto l'e&longs;ercito, che da <lb/>po&longs;&longs;anza diuina fo&longs;&longs;e gouernato in quella dife&longs;a, & mutò la ragione del <lb/>guerreggiare, dando&longs;i all'aßedio, & al vietare &longs;tret tißimamente le vitto­<lb/>uaglie a quella città.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Que&longs;te furono le cagioni, che la Mechanica &longs;alì in tanta gloria, & che <lb/>i Romani le a&longs;&longs;egnarono dapoi grado honoreuolißimo ne gli e&longs;erciti lo­<lb/>ro, come &longs;i legge nel primo libro della guerra ciuile, che Ce&longs;are &longs;e prigione <lb/>il Capitano de' fabri di Pompeio, nomato Magio Cremona, & Vitruuio fu <lb/>Capitano delle Bali&longs;te di Ce&longs;are Augu&longs;to, che &longs;arebbe nella militia moder­<lb/>na, come Capitano generale dell' artiglieria. </s> <s>La qual gloria &longs;ucceßiua­<lb/>mente le fu mantenuta poi da molti dottißimi &longs;crittori, & mae&longs;tri di <lb/>Mechani a, come da Cte&longs;ibio Ale&longs;&longs;andrino, da Herone Ale&longs;&longs;andrino, da <lb/>un'altro Herone, da Ateneo, da Bione, da Pappo Ale&longs;&longs;andrino, che allega <lb/>Carpo di Antiochia, da Eliodoro, da Oriba&longs;io, & da altri Greci, i quali fio <lb/>rirono in diuer&longs;i tempi, in&longs;egnando la ragione, la mi&longs;ura, & l'v&longs;o de gli <lb/>i&longs;trumenti bellico&longs;i non &longs;olo, ma di tutti gli altri, che le pertengono. </s> <s>Fra <lb/>Latini antichi Varrone &longs;cri&longs;&longs;e dell' Architettura, & per con&longs;eguente douet <lb/>te anco far mentione della Mechanica: & Vitruuio, & Vegetio, & qual­<lb/>che altro hanno fauellato d'intorno alla fabrica delle machine militari, <lb/>& da mouer pe&longs;i, & aiutato à con&longs;eruare fra gli huomini viua la digni­<lb/>tà della Mechanica.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma ruinando l'Imperio di Romani, & &longs;uccedendo i barbari in Italia, <lb/>in Grecia, in Egitto, & in ogni contrada, oue &longs;i e&longs;er cita&longs;&longs;ero le buone lette <lb/>re, caddero mi&longs;erabilmente, & &longs;i perderono qua&longs;i del tutto le &longs;cienze, & in <lb/>&longs;pecialità re&longs;tò la Mechanica lunghißimo tempo negletta, non cono&longs;cen­<lb/>do&longs;i in guerra altri difici, che Bricole, Trabucchi, Mangani, Martinelli, & <lb/>certi i&longs;trumenti tali, finche &longs;ouragiun&longs;e l'artiglieria, laquale à poco à po­<lb/>co gli fe di&longs;u&longs;are à fatto: & di quella parte altre&longs;i della Mechanica, laqua <lb/>le s'adopra al mouer pe&longs;i, ben picciolo intendimento rima&longs;e. </s> <s>Vera co&longs;a è, <lb/>che &longs;embra da vn tempo in quà le arti, & le dottrine più nobili, come le <lb/>belle lettere appellate humane, la Filo&longs;ofia, la Medicina, l'A&longs;trologia, l'A­<lb/>rithmetica con la Mu&longs;ica, la Geometria, l'Architettura, la Scoltura, la Pit <lb/>tura con molte altre: & &longs;pecialmente la Mechanica e&longs;&longs;ere dalle o&longs;cure te-<emph.end type="italics"/> <pb/><emph type="italics"/>nebre, oue giaccudno &longs;epolte, alla thiara luce ri&longs;u&longs;citate: Percioche ri­<lb/>&longs;tringendomi alle Mechaniche Giordano, che &longs;<gap/>ri&longs;&longs;e de' pe&longs;i, la incominciò <lb/>à &longs;olleuare alquanto, & poi Leon Batti&longs;ta Alberti nella &longs;ua Architettu­<lb/>ra: il Tartaglia aper&longs;e anco la via à molte &longs;peculationi Mechaniche: Vitto <lb/>rio Fau&longs;to nell' Arzanà di Vene<gap/>ia m<gap/>&longs;trò d'e&longs;&longs;ere buon Mechanico: Mon­<lb/>&longs;ig. </s> <s>Reuerendi&longs;s. </s> <s>barbaro eletto d' Aquileia ne' Commentari del decimo di <lb/>Vitruuio nominò gli i&longs;trumenti da mouer pe&longs;i: Georgio Agricola nel &longs;e&longs;to <lb/>de' Metalli raccol&longs;e a&longs;&longs;ai machine da leuar pe&longs;i, & qualched'vn'altro: & <lb/>nuoua <expan abbr="m&etilde;te">mente</expan> l'Autoredi que&longs;t'opera, ilquale ben d'altra maniera in ciò pro <lb/>ce lette, che gli autori nominati, peroche con ordine ammir abile, & con <lb/>vere, & certe ragioni ha in&longs;egnato &longs;olo fra Latini ottimamente que&longs;tæ <lb/>&longs;cienza tutta da mouer pe&longs;i.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma &longs;i come i moderni da me ricordati, & principalmente l'Autore del <lb/>pre&longs;ente libro hanno ornata & e&longs;altata la Mechanica con le parole, & co i <lb/>volumi; co&longs;i V. S. Illu&longs;triß. </s> <s>l'hà celebrata, & magnificata co' di&longs;cor&longs;i, & <lb/>con le operationi i&longs;te&longs;&longs;e, & co' fattire&longs;a famigliare, & dome&longs;tica, diuer&longs;e <lb/>machine fabricando con profondißima dottrina, & facendone e&longs;perten­<lb/>ze nel mouere qualunque gran pe&longs;o, di cui &longs;i po&longs;&longs;a l'huomo in ogni bi&longs;ogno <lb/>&longs;eruire. </s> <s>Talche ben &longs;i puote con verità affermare, che per vna pa<gap/>te e&longs;&longs;a, <lb/>& l'Autore di que&longs;ti trattati per l'altra, habbiate alla Mechanica il pri&longs;ti <lb/>no honore re&longs;tituito, che da i tempi antichi in quà le cra &longs;marrito.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>S<gap/>no &longs;or&longs;e quaran a anni gia &longs;cor&longs;i, che per i&longs;cherzare con Nicolò Tar <lb/>taglia, per&longs;ona à &longs;uoi tempi molto &longs;timata in que&longs;ta profeßone, & che &longs;i <lb/>dile ttaua di andare &longs;oluendo que&longs;tioni &longs;ottili di Mechanica, & di Mathe <lb/>matica, & ne' &longs;uoi dialoghi introduceua à fauellare per&longs;onaggi grandi: <lb/>& alcuna fiata gli faceua dire qualche co&longs;a, di cui eßi prendeuano onta, <lb/>V. S. Illu&longs;triß. </s> <s>gliene propo&longs;e for&longs;e quaranta Mechaniche qua&longs;i tu<gap/>te, & <lb/>difficili: alcune delle quali egli prouò di &longs;oluere, delle altre &longs;i &longs;cusò con di <lb/>re, che à cia&longs;cheduna di loro &longs;arebbe &longs;tato me&longs;tieri vn volume intero, co­<lb/>me &longs;i legge ne' &longs;uoi lib i &longs;tampati della noua &longs;cientia.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Hor non è punto di marauiglia, che ella habbia penetrato con l'inten­<lb/>dimento <expan abbr="tãto">tanto</expan> dentro, & &longs;aputo co&longs;i bene operare nelle Mechaniche, & &longs;ia <lb/>fatta padrona in tut<gap/>o dell'arte del fortificare i &longs;iti, & d'ogni altra parte <lb/>della militia: peroche fu dall'ottimo &longs;uo padre alleuata in compagnia di <lb/>huomini &longs;cientiati, & d'alio affare, tra quali fu vn tempo Con&longs;tantino <lb/>La&longs;cari nobilißimo huomo Greco, & piono di dottrina, da cui &longs;ucceßi­<lb/>uamente imparò, oltra le altre lettere, Arithmetica, Geometria, A&longs;tro-<emph.end type="italics"/> <pb/><emph type="italics"/>logia, Geogra&longs;ia; à ài&longs;egnare, & lauorare manualmente in me&longs;tieri diuer­<lb/>&longs;i; à caualcare, à maneggiare le arme, à tirare d'archibugio, & d'artiglie <lb/>ria, & à <expan abbr="cõporre">comporre</expan> fochi artificiati, & l'arte per eccellenza detta del bom <lb/>bardiero; à viuere &longs;obriamente, & le fatiche rolerare al caldo, al freddo, <lb/>& ad ogni di&longs;agio, co&longs;e tutte, che di&longs;pongono l'animo, & indurano il corpo <lb/>alla militia. </s> <s>Giunta poi all' età di &longs;edici anni, fu inutata con dodici caual <lb/>li qua&longs;i tutti Turchi, & con prouedimento conuene<gap/>ole di denari à vede­<lb/>re tutta quella guerra, che pa&longs;sò in Italia dalla pre&longs;ura del Rè France&longs;co <lb/>Primo di Francia, fin alla pace generale, che &longs;eguì l'anno<emph.end type="italics"/> 1529. <emph type="italics"/>Nella­<lb/>quale interuennero qua&longs;i tutti i mouimenti militari, che &longs;i po&longs;&longs;ano ima­<lb/>ginare, sì per gli e&longs;erciti grandi, che erano à front<gap/> l'vn contra l'altro, sì <lb/>per la qualità, & quantità delle impre&longs;e fatte, & per mille altri acciden <lb/>ti importantißimi, & &longs;tratagemi auenuti, & sì principalmente; percio­<lb/>che nell'vn campo, & l'altro in varie &longs;tagioni militarono i primi guer­<lb/>rieri del mondo, & in gran numero, i quali con prudenza, a&longs;tutia, & <lb/>brauura contendeuano à gara, & per honore di &longs;oura&longs;tare, & e&longs;&longs;ere vinci <lb/>tori. </s> <s>Et veramente chi ben con&longs;idera, fin da i tempi antichi, rarißime vol <lb/>te è &longs;tato con numero maggiore di Capitani famo&longs;i, ò con più copia d'im­<lb/>pre&longs;e grandi guerreggiato, che in quegli anni: Peroche furono fatti prigio­<lb/>ni due de' maggiori Prencipi del mondo, &longs;i a&longs;&longs;ediò Milano, & per forza fu­<lb/>rono pre&longs;e tre città, Roma, Cremona, & Pauia; &longs;i videro più fatti d'ar­<lb/>me, & gli e&longs;erciti &longs;i andarono per&longs;eguitando da Milano à Roma; &longs;i che Pia <lb/>cenza, Parma, Bologna, & Fiorenza guardaròn&longs;i dalle armi nemiche.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Nello &longs;plendore dunque della &longs;cola del Duca France&longs;co Maria d'V rbino, <lb/>ilquale era Capitano generale della Lega, & di quegli altri valentißim<gap/><lb/>Capitani, andaua V. S. Illu&longs;triß. </s> <s>come di &longs;ua libertà, & benißimo à ca­<lb/>uallo, con chi le piaceua, & &longs;i trouaua à quelle fattioni, che volea, &longs;eguen <lb/>do le più volte il Sig. </s> <s>Giouanni de' Medici, & Paulo Luzza&longs;co, che erano <lb/>&longs;empre de&longs;ti, & arditi, & come l'occhio dell'e&longs;ercito. </s> <s>Quì non è mia in­<lb/>tentione di narrare gli auenimenti di quella guerra, ma &longs;i bene di auerti <lb/>re, che chi la vide, & appre&longs;e da buon &longs;enno i &longs;uoi moti; & &longs;eppe manda­<lb/>re à memoria quei &longs;atti marauiglio&longs;i, ben puote meritamente vantar&longs;i <lb/>di hauer mirato ca&longs;i memorabili, i quali nè anche in migliaia d'anni &longs;o­<lb/>gliono accadere; come ella, che e&longs;&longs;endo giouine di viuace &longs;pirito, & am­<lb/>mae&longs;trata nclle arti nece&longs;&longs;arie al&longs;oldato, & volentern&longs;ißima d'imparare, <lb/>hebbe opportu<gap/>a occa&longs;ione di far&longs;i prattica d<gap/>ll'ordinare, de<gap/>'e&longs;ercitare, <lb/>del far marciare in battaglia, dell'alloggiare in campagna gli e&longs;erciti &longs;i-<emph.end type="italics"/> <pb/><emph type="italics"/>hauendone ella fabricate di &longs;ua indu&longs;tria for&longs;e dodici di maniere differen <lb/>ti, parte da &longs;tra &longs;cinare, & parte da alzare con pochißima forza &longs;mi&longs;u­<lb/>rati pe&longs;i: come quella, che hà vna &longs;ola rota co' denti, & ali'erta tira cin­<lb/>que de' &longs;uoi canoni con la po&longs;&longs;anza di Grada&longs;&longs;o &longs;uo Nano: & quell'altra, la <lb/>quale con vna oncia di forza &longs;ola, po&longs;ta nel manico, che la volge, dà il me <lb/>to à quattordici mila libre di pe&longs;o: che &longs;e al detto manico &longs;i attribui&longs;ce <lb/>la forza, che comunalmente haue l'huomo con la mano, cioè libre cinquan <lb/>ta, egli è manife&longs;to la predetta machina hauere po&longs;&longs;anza di mouere, co&longs;a <lb/>incredibile, molto più di otto millioni di libre. </s> <s>Que&longs;te machine portabi­<lb/>li da vn mulo, & alcune anche da vn'huomo &longs;ono à diuer&longs;i affari nece&longs;&longs;a­<lb/>rijßime, & maßimamente à maneggiare, & condurrei pezzi großi del­<lb/>l'artiglieria. </s> <s>& per certo &longs;e l'anno<emph.end type="italics"/> 1529. <emph type="italics"/>il Conte di San Polo Capitano <lb/>France&longs;e nel ritirar&longs;i dall'a&longs;&longs;edio di Milano inuer&longs;o Piemonte con l'e&longs;er­<lb/>cito, & con l'artiglieria, haue&longs;&longs;e portato &longs;eco vno de' minimi i&longs;trumenti <lb/>d'O&longs;opo, non &longs;arcbbe &longs;cor&longs;o in quello &longs;tremo infortunio, percioche in mar <lb/>ciando fu da vn graue canone rotto il ponte, che trauer&longs;aua il &longs;o&longs;&longs;o della <lb/>&longs;trada, & il pezzo cadè nel fango. </s> <s>Onde formoßi il campo per non la­<lb/>&longs;ciarlo à dietro, & non hauendo ingegno da cauarlo fuori, &longs;i con&longs;umò tan­<lb/>to tempo, che &longs;opragiun&longs;e Antonio da Leua con le &longs;ue genti, & ritrouan <lb/>do l'e&longs;&longs;ercito nemico &longs;eparato, & in quel di&longs;ordine, lo mi&longs;e in rotta, & fè <lb/>preda delle bagaglie, delle artiglierie, & del Capitano mede&longs;mo. </s> <s>Non hà <lb/>troppo iempo, che il Duca France&longs;co di Gui&longs;a, allhor che di Francia guidò <lb/>l'e&longs;ercito in Abruzzo, douendo partire, volle &longs;piegare prima la fanteria, <lb/>& cauælleria &longs;ua in ordinanza à fronte del nemico, qua&longs;i à battaglia sfi­<lb/>dandolo; ma poi nel ritorno &longs;caualco&longs;si vn pezzo d'artiglieria, & s'arre­<lb/>&longs;tò tutta la ma&longs;&longs;a delle genti, & quei Prencipi France&longs;i &longs;montati da ca­<lb/>uallo, penarono buona pezza auanti, che lo ripone&longs;&longs;ero &longs;u le rote, con ri&longs;chio <lb/>di patir danno da gli aduer&longs;ari, che haue&longs;&longs;ero con quella occa&longs;ione &longs;pinto <lb/>innanzi. </s> <s>Di quc&longs;ti e&longs;empi non mancano per l'hi&longs;torie.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Hora che è pace V. S. Illu&longs;triß. </s> <s>è andata inue&longs;tigando per &longs;uo diporto <lb/>molte, & varie &longs;orti di ordigni da mouer pe&longs;i, affine di valer&longs;ene nelle <lb/>fabriche, & nell'argine di pietre, che fa per ritenere l'impeto del Taglia­<lb/>mento, che n<gap/>n gua&longs;ti i colti di O&longs;opo, & per douer&longs;ene anco &longs;eruire, quan <lb/>do che &longs;ia in guerra. </s> <s>Si come fece Archimede, ilquale, &longs;econdo Plutarco, &longs;tan <lb/>do in pace à petitione di Hierone Rè, compo&longs;e quelle <expan abbr="tãie">tanie</expan> Machine per giuo­<lb/>co, & i&longs;ch<gap/>rzo di Geometria, l<gap/>quali poi &longs;oprauenendo la guerra, le &longs;eppe <expan abbr="cõ">com</expan> <lb/>uertire opportunamente contra Romani. </s> <s>Et &longs;e egli, come te&longs;tificano diuer&longs;i<emph.end type="italics"/> <pb/><emph type="italics"/>autori, &longs;edendo con certa machina detta, &longs;econdo Oriba&longs;io, Tri&longs;pa&longs;ton, per <lb/>che &longs;i maneggiaua con tre cor de, tirò dal mare in terra quella gran naue <lb/>del Rè &longs;uo; & con la forza della mano &longs;ini&longs;tra mo&longs;&longs;e mediante l'i&longs;tru­<lb/>mento vn pe&longs;o di cinque mila &longs;taia ò moggia, sì fattamente che diputan­<lb/>do à cia&longs;cuno &longs;taio quarantacinque libre di pe&longs;o, a&longs;cenderebbono alla &longs;om­<lb/>ma di dugento venticinque mila libre; & pre&longs;umeua&longs;i di hauer potuto <lb/>mouere la terra, trouando doue fermar&longs;i con la leua, ò con quella &longs;ua ma­<lb/>china de&longs;critta da Pappo nell'ottauo libro delle raccolte matematiche, la <lb/>quale hauea cinque rote co' &longs;uoi as&longs;i, & vna vite perpetua co'l manico: Io <lb/>mi rendo certo, che ella s'ingegnerebbe di &longs;ormare i&longs;trumenti per adoprare <lb/>altretanto.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Hauendo io dunque veduti, & i&longs;perimentati que&longs;ti vari difici ad O&longs;o­<lb/>po; & es&longs;endomi &longs;tato da lei mo&longs;trato la prima volta il pre&longs;ente libro, & <lb/>commendato &longs;ommamente, mi propo&longs;i nell' animo, che vtile &longs;arebbe il ri <lb/>durlo in volgare, accioche coloro i quali &longs;ono atti per altro ad intenderlo, <lb/>ma non hanno cono&longs;<gap/>enza del Latino, pote&longs;sero, farne &longs;uo profitto. </s> <s>Co&longs;i <lb/>compiuta l'opera, & fattala &longs;tampare, la mando à V. S. Illu&longs;tri&longs;s. </s> <s>che po&longs; <lb/>&longs;ede e&longs;qui&longs;it amente que&longs;ta materia, & &longs;econda i &longs;tudi delle buone lette­<lb/>re, i quali, &longs;e dopo Iddio, non vengono fauoriti da i gran Signori, nulla va <lb/>gliono. </s> <s>Che &longs;e in qualche parte haurò à gli amatori delle Mechani<gap/>he re­<lb/>cata ageuolezza, & vtilità con le mie fatiche, douranno eglino &longs;aper' à <lb/>lei buon grado, che di que&longs;ta fattura è &longs;tata cagione. </s> <s><lb/>Di Venetia à<emph.end type="italics"/> 28. <emph type="italics"/>di Giugno<emph.end type="italics"/> 1581. </s> </p> <p type="main"> <s><emph type="italics"/>Di V. S. Illu&longs;tri&longs;s.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Affettionatiß. </s> <s>&longs;cruidore <lb/>Filippo Pigafetta.<emph.end type="italics"/></s> </p> <pb/> <p type="head"> <s>AI LETTORI</s> </p> <p type="main"> <s>Il prefente libro contiene &longs;ei trattati, il primo de <lb/>quali è della Bilancia con la Stadera, l'altro della <lb/>Leua, il terzo della Taglia, il quarto dell' A&longs;&longs;e nel­<lb/>la rota, il quinto del Cuneo, & l'vltimo della Vite, <lb/>che tutti &longs;ono i&longs;trumenti Mechanici. </s> <s>Intitula&longs;i le <lb/>Mechaniche. </s> <s>Ma percioche que&longs;ta parola Mechaniche non ver <lb/>rà for &longs;e int e&longs;a da cia&longs;cheduno perlo &longs;uo vero &longs;ignificato, anzi <lb/>troueran&longs;i di quelli, che &longs;timeranno lei e&longs;&longs;ere voce d'ingiuria, <lb/>&longs;olendo&longs;i in molte parti d'Italia dire ad altrui Mechanico per <lb/>i&longs;cherno, & villania; & alcuni per e&longs;&longs;ere chiamati Ingegnieri &longs;i <lb/>prendono &longs;degno: non &longs;arà per auentura fuori di propo&longs;ito il <lb/>ricordare, che Mechanico è vocabolo honorati&longs;&longs;imo, dimo&longs;tran <lb/>te, &longs;econdo Plutarco, me&longs;tiero alla Militia pertinente, & conue <lb/>neuole ad huomo di alto affare, & che &longs;appia con le &longs;ue mani, <lb/>& co'l &longs;enno mandare ad e&longs;ecutione opre marauiglio&longs;e à &longs;ingu <lb/>lare vtilità, & diletto del viuere humano. </s> </p> <p type="main"> <s>Fù, per nomiuarne alcuno tra molti Filo&longs;ofi, & Prencipi de' <lb/>preteriti &longs;ecoli, Archita Tarentino, & Eudo&longs;&longs;o <expan abbr="cõpagni">compagni</expan> di Pla­<lb/>tone, & valenti&longs;&longs;imi Ingegnieri, & Mechanici, che &longs;ono vna me <lb/>de&longs;ma co&longs;a, di cui fa Plutarco mentione nella vita di Marcello: <lb/>& Demetrio Rè, inuentore &longs;ottili&longs;&longs;imo di Machine bellico&longs;e, <lb/>& ne lauoraua di &longs;ua mano ancora: & fra Greci di Sicilia Me­<lb/>chanico, & Ingegniere famo&longs;is&longs;imo Archimede Siracu&longs;ano, il <lb/>quale era di <expan abbr="grã">gram</expan> legnaggio, & parente di Hierone Rè di Sicilia. </s> </p> <p type="main"> <s>Et quantunque Plutarco nell'i&longs;te&longs;&longs;a vita affermi, che egli di <lb/>&longs;pregia&longs;&longs;e le Mechaniche, come bas&longs;i & vili, & materiali, nè di <lb/>loro degna&longs;&longs;e &longs;criuere giamai, & che non per opera principale, <lb/>ma per vn cotale &longs;ollazzo, & giuoco di Geometria impiegaua <lb/>la fatica nelle Mechaniche, pregato da quel Rè; sì leggiamo <lb/>noi tuttauia in altri autori, lui hauere dettato vn libro della mi <lb/>&longs;ura, & proportione d'ogni maniera di va&longs;ello, diui&longs;ando la for <lb/>ma della gran naue fabricata da Hierone, à cui nulla manca­<lb/>ua: & Pappo Ale&longs;&longs;andrino allega il libro della Bilancia di Ar­<lb/>chimede, che è pur Mechanico tutto: & l'i&longs;te&longs;&longs;o nell'ottauo del <lb/>le raccolte Matematiche pone vn'i&longs;trumento da mouer pe&longs;i, <pb/>tentato l'Autore di manife&longs;tare per hora, & il primo de Latini <lb/>con dimo&longs;trationi ageuoli, & piane, in&longs;egnare &longs;olamente la ra­<lb/>gion dello intendere, & maneggiare gli &longs;ei predetti l&longs;trumenti <lb/>Mechanici; à quali &longs;i riducono tutti gli altri, come à &longs;uoi prin­<lb/>cipil, & fondamenti; & da'quali &longs;i po&longs;&longs;ono comporne diuer&longs;e ma <lb/>niere, accozzandone in&longs;ieme due, tre, & più, come l'A&longs;&longs;e nella <lb/>rota con la Taglia, la Vite co'l detto A&longs;&longs;e, & con la Leua, & &longs;uc­<lb/>ces&longs;iuamente de gli altri ad arbitrio di chiunque in varie opre &longs;e <lb/>ne sà con giudicio valere, come nota l'Autore nel fine di que&longs;to <lb/>volume. </s> </p> <p type="main"> <s>Hor come che l'Autore con bella via, & chiara, & con ordine <lb/>ammirabile di que&longs;ti difici habbia ragionato, & la co&longs;a per &longs;e <lb/>molto o&longs;cura non &longs;ia ad intender&longs;i: nondimeno ben ricerca ella <lb/>tutto l'intelletto dell'huomo, & che con &longs;i&longs;&longs;a &longs;peculatione &longs;i leg­<lb/>gano attentis&longs;imamente più d'vna volta le dimo&longs;trationi. </s> </p> <p type="main"> <s>Doue &longs;i vede in alcuni luoghi di que&longs;ti trattati cotale &longs;orte di <lb/>lettere picciole, differente dalle altre, come la pre&longs;ente; auer­<lb/>ta&longs;i che non vi &longs;ono co&longs;e dettate dall' Autore di que&longs;to libro di <lb/>Mechaniche, ma notate da colui che l'hà volgarizato, à fine di <lb/>chiarire qualche pa&longs;&longs;o difficile, & ageuolare l'intendimento à' <lb/>Lettori non co&longs;i prattichi nelle Scole de' Filo&longs;ofi. </s> </p> <p type="main"> <s>Ponga&longs;i anco mente, che à carte 121. nel trattato della Vite, <lb/>è po&longs;to fra i detti dell' Autore il Problema di Pappo, ilquale do­<lb/>uea e&longs;&longs;ere &longs;tampato con lettere differenti dalle altre, ma per in­<lb/>auertenza è &longs;tato me&longs;&longs;o co' caratteri &longs;te&longs;&longs;i delle propo&longs;itioni del <lb/>l' Autore, che è difetto. </s> <s>Non è &longs;tato pos&longs;ibile &longs;chluare alcuni <lb/>falli nello &longs;tampare. </s> <s>Onde correggan&longs;i in que&longs;ta maniera. </s> <s>Nel <lb/>la Lettera à carte 1. faccia 2. ver&longs;i 25. to&longs;&longs;enoni, leggi tollenoni. </s> <s><lb/>car. </s> <s>43. ver. </s> <s>22. dell'angolo, all'angolo. </s> <s>carte 48. f. </s> <s>2. nella po­<lb/>&longs;tilla, per la 2. di que&longs;to; della 2. di que&longs;to. </s> <s>carte 87. f. </s> <s>2. <lb/>ver. </s> <s>14. dalla, alla. </s> <s>carte 93. ver. </s> <s>32. cni, cui. </s> <s>carte 115. ver. </s> <s>20. <lb/>Hlici, Helici. </s> <s>Gli altri errori di lettere meno importanti, & che <lb/>non mouono il &longs;en&longs;o alla, di&longs;cretione del giudicio&longs;o Lettore &longs;i ri <lb/>mettono. </s> </p> <pb/> <p type="main"> <s>mo&longs;trando e&longs;&longs;ere il quarante&longs;imo trouato d' Archimede, per cui <lb/>di&longs;&longs;e; Dami oue io mi fermi, ch'io mouerò la terra; & Carpo <lb/>Mechanico &longs;c ri&longs;&longs;e, che Archimede compo&longs;e vn libro del modo <lb/>del fare le Sfere, che è fattura Mechanica. </s> <s>Ma più il mede&longs;imo <lb/>Archimede, non vna &longs;ola volta cita &longs;e &longs;te&longs;&longs;o, nel libro della Qua <lb/>dratura della Parabola, con parole tali. </s> <s>Imperoche egli è dimo­<lb/>&longs;trato nelle Mechaniche; accennando alcune propo&longs;itioni del <lb/>&longs;uo libro delle co&longs;e, che egualmente pe&longs;ano, ilquale è tutto Me­<lb/>chanico. </s> <s>Oltre à ciò vna parte del libro della Quadratura della <lb/>Parabola, & il &longs;econdo delle co&longs;e, che &longs;tanno &longs;opra l'acqua, oue­<lb/>ro à galla &longs;ono Mechanici. </s> <s>Da que&longs;ti luoghi vede&longs;i e&longs;pre&longs;&longs;o, che <lb/>non &longs;olamente Archimede fece opre Mechaniche, ma ne &longs;cri&longs;&longs;e <lb/>anco molti trattati; & confe&longs;&longs;a Plutarco per niuna altra dottri­<lb/>na e&longs;&longs;ere tanto in riputatione &longs;alito Archimede, quanto per le <lb/>impre&longs;e Mechaniche; anzi veramente co'l mezo loro hauer&longs;i egli <lb/>all'hora procacciato fama non di &longs;cienza humana, ma di &longs;apien­<lb/>za diuina. </s> <s>Per la qual co&longs;a egli è ben da con&longs;iderare, come Plu­<lb/>tarco &longs;i &longs;ia la&longs;ciato tra&longs;correr' à dire, che Archimede le Mechani <lb/>che di&longs;preggia&longs;&longs;e, nè di loro degna&longs;&longs;e &longs;criuere: & per certo egli <lb/>forte d'opinione &longs;arebbe&longs;i <expan abbr="ingãnato">ingannato</expan>, &longs;e haue&longs;&longs;e poco &longs;timata quel <lb/>la facultà, che lo fè guadagnare gloria di gran lunga maggio­<lb/>re, che qualunque altra &longs;cienza &longs;i po&longs;&longs;ede&longs;&longs;e. </s> <s>Vitruuio de i <lb/>Latini fù buon Mechanico, & &longs;eruì per Capitano delle Bali&longs;te, <lb/>& delle altre machine da guerra Ottauiano Ce&longs;are, & gli intitu­<lb/>lò le &longs;ue fatiche dell' Architettura, & ne diuenne ricco. </s> </p> <p type="main"> <s>L'e&longs;&longs;ere Mechanico dunque, & Ingegniero con l'e&longs;empio di <lb/>tanti valent'huomini, è officio da per&longs;ona degna, & &longs;ignorile: <lb/>& Mechanica è voce Greca &longs;ignificante co&longs;a fatta con artificio <lb/>da mouere, come per miracolo, & fuori dell'humana po&longs;&longs;anza <lb/>grandis&longs;imi pe&longs;i con picciola forza, & in generale comprende <lb/>cia&longs;cun Dificio, Ordigno, I&longs;trumento, Argano, Mangano, oue­<lb/>ro ingegno mae&longs;treuolmente ritrouato, & lauorato per cotali ef <lb/>fetti, & &longs;imili altri infiniti in qual &longs;i voglia &longs;cienza, arte, & e&longs;er­<lb/>citio. </s> <s>Laquale hò de&longs;critta co&longs;i materialmente per darne vn cer <lb/>to &longs;aggio accommodato al gu&longs;to del più de gli huomini; trala­<lb/>&longs;ciando le accurate di&longs;&longs;initioni à miglior tempo. </s> </p> <p type="main"> <s>Aggiunga&longs;i, che &longs;otto que&longs;to vniuer&longs;ali&longs;&longs;imo titolo &longs;i è con-</s> </p> <pb pagenum="1"/> <p type="head"> <s>LIBRODI <lb/>MECHANICHE, <lb/>DELL'ILLVSTRISSIMO <lb/>SIGNORE, <lb/>II. S. GVIDO VBALDO DE' MARCHESI <lb/>DEL MONTE.</s> </p> <figure></figure> <p type="head"> <s>Diffinitioni.</s> </p> <p type="main"> <s>Il centro della grauezza di cia&longs;cun corpo e vn <lb/>certo punto po&longs;to dentro, dal quale &longs;e con la <lb/>imaginatione s'intende e&longs;&longs;erui appe&longs;o il gra­<lb/>ue, mentre è portato &longs;ta fermo, & mantiene <lb/>quel &longs;ito, che egli hauea da principio, ne in <lb/>quel portamento &longs;i và riuolgendo. </s> </p> <p type="main"> <s><emph type="italics"/>Que&longs;ta diffinitione del centro della grauezza in&longs;egnò <lb/>Pappo Ale&longs;&longs;andrino nell'ottauo libro delle raccolte ma­<lb/>thematiche. </s> <s>Ma Federico Comandino nel libro del cen­<lb/>tro della grauezza de' corpi &longs;olidi dichiarò l'i&longs;te&longs;&longs;o centro in questa maniera de&longs;cri­<lb/>uendolo.<emph.end type="italics"/></s> </p> <p type="main"> <s>Il centro della grauezza di cia&longs;cuna figura &longs;olida è quel punto po&longs;to <lb/>dentro, d'intorno alquale le parti di momenti eguali da ogni parte <lb/>&longs;i fermano. </s> <s>Peroche &longs;e per tale centro &longs;arà condotto vn piano, che <lb/>&longs;eghi in qual &longs;i voglia modo la figura, &longs;empre la diuiderà in parti, <lb/>che pe&longs;eranno egualmente. </s> </p> <pb/> <p type="head"> <s>NOTITIE COMVNI.</s> </p> <p type="head"> <s>I.</s> </p> <p type="main"> <s>Se da co&longs;e egualmente pe&longs;anti &longs;i leneranno co&longs;e, che pur egualmente <lb/>pe&longs;ino, le re&longs;tanti pe&longs;eranno egualmente. </s> </p> <p type="head"> <s>II.</s> </p> <p type="main"> <s>Se à co&longs;e egualmente pe&longs;anti &longs;i aggiungeranno co&longs;e, che pur <expan abbr="egualm&etilde;">egualmen</expan> <lb/>te pe&longs;ino, tutte in&longs;ieme pe&longs;eranno egualmente. </s> </p> <p type="head"> <s>III.</s> </p> <p type="main"> <s>Le co&longs;e, che all'i&longs;te&longs;&longs;o &longs;ono eguali in pe&longs;o, &longs;ono tra loro anco gra­<lb/>ui egualmente. </s> </p> <p type="head"> <s>PRESVPPOSTE.</s> </p> <p type="head"> <s>I.</s> </p> <p type="main"> <s>Di vno corpo è vn &longs;olo centro della grauezza. </s> </p> <p type="head"> <s>II.</s> </p> <p type="main"> <s>Il centro della grauezza di vn corpo è &longs;empre nel mede&longs;imo &longs;ito per <lb/>ri&longs;petto al &longs;uo corpo. </s> </p> <p type="head"> <s>III.</s> </p> <p type="main"> <s>I Pe&longs;i &longs;ono portati in giu &longs;econdo il centro della grauezza. </s> </p> <p type="main"> <s>DIFFINITIONI. </s> <s>La diffinitione è vn breue parlare, che manife&longs;ta, & inte­<lb/>ramente dichiara la co&longs;a propo&longs;ta, &longs;i fattamente che non &longs;i po&longs;&longs;a trouare condi­<lb/>tione, ouero accidente alcuno principale in e&longs;&longs;a co&longs;a, &longs;e la diffinitione è buona, <lb/>che non &longs;ia in virtù compre&longs;a, & detta da lui; come per e&longs;empio l'Autore qui di <lb/>&longs;opra da ad inten dere che &longs;ia il centro della grauezza con due diffinitioni. </s> </p> <p type="main"> <s>Le Notitie comuni poi &longs;ono certe &longs;entenze manife&longs;te al &longs;en&longs;o comune de gli huomi­<lb/>ni, lequali pur che vi &longs;i ponga mente, &longs;ubito vdite, &longs;i intendono, & &longs;e le pre&longs;ta il <lb/>con&longs;entimento. </s> </p> <p type="main"> <s>Ma la Pre&longs;uppo&longs;ta è diuer&longs;a, peroche mette per vero la co&longs;a co&longs;i e&longs;&longs;ere, come &longs;i pro­<lb/>pone &longs;enza altro di&longs;cor&longs;o per farla cono&longs;cere. </s> </p> <pb/> <p type="head"> <s>TRATTATI IN QVEST'OPERA <lb/>CONTENVTI.<lb/> </s> </p> <table><row><cell>I.</cell><cell>Della Bilancia, con la Stadera à carte</cell><cell>1</cell></row><row><cell>II.</cell><cell>Della Leua.</cell><cell>35</cell></row><row><cell>III.</cell><cell>Della Taglia.</cell><cell>56</cell></row><row><cell>IIII.</cell><cell>Dell' A$$e nella Rota.</cell><cell>102</cell></row><row><cell>V.</cell><cell>Del Cuneo.</cell><cell>107</cell></row><row><cell>VI.</cell><cell>Della Vite.</cell><cell>115</cell></row></table> <pb pagenum="2"/> <p type="head"> <s>DELLA BILANCIA</s> </p> <p type="main"> <s>Avanti che &longs;i faccia mentione della Bilancia, accioche la <lb/>co&longs;a re&longs;ti più chiara, &longs;ia la Bilancia AB in linea diritta, & <lb/>CD la Truttina della Bilancia, laquale &longs;econdo la con&longs;uetu <lb/>dine comune &longs;tà &longs;empre à piombo dell'orizonte. </s> <s>& il punto C im <lb/>mobile, d'intorno alquale &longs;i volge la Bilancia, &longs;i chiami il centro del <lb/>la bilancia, &longs;ia pur collo­<lb/>cato di &longs;opra della bilan <lb/>cia, ò di &longs;otto, benche <lb/>non propriamente, che <lb/>non fa nulla Ma il CA, <lb/>& il CB &longs;iano le di&longs;tan <lb/>ze, & braccia della Bilan <lb/>cia, co&longs;i nomate. </s> <s>& &longs;e <lb/>dal centro della bilancia <lb/>collocato di &longs;opra, ò di <lb/>&longs;otto della Bilancia, &longs;arà <lb/>tirata vna linea à piom­<lb/>bo di AB, que&longs;ta &longs;i chia <lb/>merà perpendicolo, che <lb/>&longs;o&longs;terrà la Bilancia AB, <lb/>& &longs;empre &longs;tarà à piom­<lb/>bo di e&longs;&longs;a Bilancia, mo­<lb/>ua&longs;i ella in qual &longs;i voglia <lb/>modo. </s> </p> <figure></figure> <p type="main"> <s>Concio&longs;ia che in que&longs;to trattato della Bilancia, & ne gli altri ancora l'Autore v&longs;i <lb/>alcune parole, lequali non &longs;i &longs;ono potute tra&longs;portare commodamente in v olga­<lb/>re, per non e&longs;&longs;ere e&longs;&longs;e anco &longs;tate accettate in que&longs;ta lingua, ne inte&longs;e da ognuno, <lb/>io le ho la&longs;ciate co&longs;i latine. </s> <s>Ma accioche non facciano difficultà à coloro, i quali <lb/>non intendono il latino, le andrò per tutto à fuoi luoghi dichiarando. </s> </p> <p type="main"> <s>Nel re&longs;to poi delle parole mi &longs;ono attenuto più al chiaro, & all'v&longs;ato, che &longs;ia pos&longs;i­<lb/>bile, & ho po&longs;to angolo retto, & linea retta in cambio di angolo diritto, & linea <lb/>diritta, & linea della direttione in lo co di linea della dirittura, & co&longs;i diretto per <lb/>diritto, & alcuna volta magnitudine in vece di grandezza, & angolo mi&longs;to per <lb/>me&longs;colato, & angolo curuilineo per di linee torte, & linea curua per torta, & &longs;oli­<lb/>do per &longs;odo, & for&longs;e qualche altro vocabolo poco v&longs;ato in que&longs;ta no&longs;tra fauella, <lb/>&longs;timando che cote&longs;te parole &longs;iano per dimo&longs;trare maggiormente la co&longs;a, & la in­<lb/>tentione dell' Autore: & etiandio de&longs;iderando, che &longs;i rendano famigliari, & dome <lb/>&longs;tiche in que&longs;ta &longs;cienza, talche ognuno le po&longs;&longs;a ageuolmente intendere. </s> </p> <p type="main"> <s>Trutina è quella co&longs;a, che &longs;o&longs;tiene tutta la Bilancia, laquale Trutina pigli a il Perno, <lb/>ouero l'A&longs;&longs;etto, & noma&longs;i in que&longs;ti pae&longs;i Gioa, altroue Giouola, ouero l'o recchie <lb/>della Bilancia, & in altre contrade Scocca, talche non &longs;i troua &longs;in hora vocabolo, <pb/>che in Italia communcmente vi &longs;i confaccia, ne alcuno di qne&longs;ti &longs;arebbe inte&longs;o <lb/>per tutto. </s> <s>Onde io ho &longs;critto co&longs;i la Trutina, &longs;perando, che &longs;i habbia à fare termi <lb/>ne, & parola generale à tutte le nationi d'Italia. </s> </p> <p type="main"> <s>Perpendicolo vuol dire quella linea, che &longs;porge in fuori dal centro della Bilancia al <lb/>mezo di detta Bilancia, ilqual Perpendicolo è &longs;olamente nelle Bilancie, lequali han <lb/>no il centro di fuori della Bilancia, o &longs;ia di &longs;otto, ò &longs;ia di &longs;opra. </s> <s>Ma quando il cen­<lb/>tro della Bilancia è nel mezo di e&longs;&longs;a, all'hora non vi è que&longs;to Perpendicolo per e&longs; <lb/>&longs;ere il centro della Bilancia, & il mezo di e&longs;&longs;a vn'i&longs;te&longs;&longs;o punto. </s> <s>Et que&longs;to Perpen­<lb/>dicolo è co&longs;a imaginata dall' Autore &longs;olamente, & non da altri, per ageuolare al­<lb/>cune dimo&longs;trationi della Bilancia, che di nouo ha inue&longs;tigate: & non è la linguet­<lb/>ta, ne meno la linea della direttione, ò dirittura che &longs;i habbia à dire. </s> </p> <p type="head"> <s>LEMMA.</s> </p> <p type="main"> <s>Sia la linca AB à piombo dell'orizonte, & col diametro AB &longs;i de&longs;cri­<lb/>ua il cerchio AEBD, il cui centro &longs;ia C. </s> <s>Dico il punto B e&longs;&longs;ere <lb/>l'infimo luogo della circonferenza del cerchio AEBD, & il pun­<lb/>to A il piu alto, & quali &longs;i voglian punti, come DE, i quali &longs;iano <lb/>però egualmente di&longs;tanti da A e&longs;&longs;ere egualmente po&longs;ti di &longs;otto, & <lb/>quelli che &longs;tanno piu da pre&longs;&longs;o ad e&longs;&longs;o A, e&longs;&longs;ere più alti di quelli, che <lb/>&longs;ono più da lunge. </s> </p> <p type="main"> <s> <arrow.to.target n="note1"></arrow.to.target><emph type="italics"/>Allunghi&longs;i la linea AB fin al centro del mondo, <lb/>che &longs;ia F. </s> <s>Dapoi &longs;ia pre&longs;o nella circonferenza <lb/>del cerchio qual &longs;i voglia punto, come G, & &longs;i <lb/>congiungano le linee FG FD FE. </s> <s>Hor per­<lb/>cioche BF è la minima linea di tutte quelle, <lb/>che dal punto F &longs;ono tirate alla circonferenza <lb/>AEBD, &longs;arà la BF minore della FG. </s> <s>Per <lb/>laqual co&longs;a il punto B &longs;arà piu da pre&longs;&longs;o al pun­<lb/>to F, che il G. </s> <s>Et per cotesta ragione &longs;i dimo­<lb/>strerà, che il punto B &longs;ta più da pre&longs;&longs;o al centro <lb/>del mondo di qual &longs;i voglia altro punto della cir­<lb/>conferenza del cerchio AEBD. </s> <s>Sarà dunque <lb/>il punto B l'infimo luogo della circon&longs;erenza del <lb/>cerchio AEBD. </s> <s>Dapoi perche AF tirata <lb/>per lo centro è maggiore di GF, &longs;arà il punto A <lb/>più alto non &longs;olamente di G, ma etiandio di qual <lb/>&longs;i voglia altro punto della circon&longs;erenza del cer­<lb/>chio AEBD. </s> <s>Oltre à ciò perche DF, & FE <lb/>&longs;ono eguali, i punti DE &longs;aranno egualmente di <lb/>stanti dal centro del mondo. </s> <s>Et e&longs;&longs;endo DF <lb/>maggiore di FG, &longs;arà il punto D, che è più da <lb/>pre&longs;&longs;o al punto A, più alto del punto G, lequali <lb/>co&longs;e tutte erano da mo&longs;trar&longs;i.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note1"></margin.target><emph type="italics"/>Per la ottaua del terzo.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="3"/> <p type="main"> <s>Que&longs;to vocabolo Lemma greco v&longs;ato da tutti i volgarizatori di Euclide, & da gli <lb/>altri Scrittori di Mathematica ancora, hò accettato anch'io. </s> <s>Ma ben con tutto ciò <lb/>&longs;timo che egli habbia me&longs;tieri di vn poco di lume per e&longs;&longs;er inte&longs;o; & viene à dire, <lb/>&longs;i come nota Cicerone nel &longs;econdo della Diuinatione, co&longs;a che prima &longs;i prende <lb/>per render facile l'intendimento delle co&longs;e, lequali &longs;i hanno dapoi à mo&longs;trare, & <lb/><expan abbr="nõ">non</expan> è Pre&longs;uppo&longs;ta, perche ella <expan abbr="nõ">non</expan> &longs;i proua <expan abbr="cõ">com</expan> ragione, ma &longs;uppon&longs;i; ma il Lemma <lb/>&longs;i dimo&longs;tra, come in que&longs;to luogo, che prende il punto B e&longs;&longs;ere po&longs;to nell'infimo <lb/>&longs;ito della circonferenza del cerchio, & lo proua per douer&longs;ene valere nelle &longs;eguen <lb/>ti dimo&longs;trationi. </s> </p> <p type="main"> <s>Doue in que&longs;to Lemma &longs;i dice, che la linea AB è à piombo dell'orizonte, intenda&longs;i <lb/>per orizonte il piano della campagna, & del terreno &longs;ottopo&longs;to, volendo dire ori <lb/>zonte parola greca vn cerchio, che termina la no&longs;tra veduta, & abbraccia & diui <lb/>de la metà della terra tutta. </s> <s>Quando dunque &longs;i troua in que&longs;ti libri vna linea, oue­<lb/>ro altra quantità e&longs;&longs;ere à piombo, ouero egualmente di&longs;tante, ò inchinata all'ori­<lb/>zonte, intenda&longs;i per l'orizonte il piano della campagna, ò del terreno. </s> </p> <p type="head"> <s>PROPOSITIONE I.</s> </p> <p type="main"> <s>Se il pe&longs;o &longs;arà &longs;o&longs;tenuto nel centro della &longs;ua grauezza da linea diritta <lb/>non &longs;i fermerà giamai, &longs;e quella i&longs;te&longs;&longs;a linea non &longs;arà à piombo del <lb/>l'orizonte. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A, & il centro della &longs;ua <lb/>grauezza B, ilqual pe&longs;o venga &longs;o <lb/>&longs;tenuto dalla linea CB. </s> <s>Dico che <lb/>il pe&longs;o non è per fermar&longs;i giamai, <lb/>&longs;e CB non &longs;arà à piombo dell'o­<lb/>rizonte. </s> <s>Sia il punto C immobi­<lb/>le, e&longs;&longs;endo co&longs;i nece&longs;&longs;ario, accio il <lb/>pe&longs;o &longs;ia &longs;o&longs;tenuto: & e&longs;&longs;endo il pun <lb/>to C immobile, &longs;e il pe&longs;o A de­<lb/>ue&longs;i mouere, il punto B de&longs;criuerà <lb/>la circonferenza di vn cerchio, il <lb/>cui mezo diametro &longs;arà CB. </s> <s>Per <lb/>laqual co&longs;a &longs;u'l centro A & con <lb/>lo &longs;patio BC &longs;i de&longs;criua il cerchio <lb/>BFDE. </s> <s>& &longs;ia di prima BC à <lb/>piombo dell'orizonte, & &longs;ia tirata <lb/>&longs;in à D, & il punto C &longs;tia di &longs;ot <lb/>to al punto B. </s> <s>Hor percioche il pe&longs;o A &longs;i moue in giù &longs;econdo il centro della gra-<emph.end type="italics"/> <arrow.to.target n="note2"></arrow.to.target><lb/><emph type="italics"/>uezza, il punto B &longs;i mouerà in giù, oue naturalmente inchina ver&longs;o il centro del mon <lb/>do per la linea diritta BD: tutto il pe&longs;o A dunque con B &longs;uo centro della gra­<lb/>uezza, grauerà &longs;opra la linea diritta BC, & concio&longs;ia che il pe&longs;o venga &longs;o&longs;tenuto <lb/>dalla linea CB, la linea CB &longs;o&longs;terrà tutto il pe&longs;o A, &longs;opra laquale non puote mo<emph.end type="italics"/> <pb/><emph type="italics"/>uer&longs;i in giù, e&longs;&longs;endogliene da e&longs;&longs;a <lb/>vietato. </s> <s>Per la diffinitione dun­<lb/>que del centro della grauezza, il <lb/>punto B & il pe&longs;o A &longs;taranno <lb/>in que&longs;to &longs;ito. </s> <s>& quantunque <gap/>il <lb/>B &longs;ia piu alto di qual &longs;i voglia al­<lb/>tro punto del cerchio, t<gap/>tauia non <lb/>&longs;i mouerà in giù da que&longs;to &longs;ito per <lb/>la circonferenza del cerchio, pero­<lb/>che non &longs;i inchinerà più ver&longs;o lo F, <lb/>che ver&longs;o lo E, per e&longs;&longs;ere nell'vna <lb/>parte & nell'altra eguale la di&longs;ce­<lb/>&longs;a: ne il pe&longs;a A piu &longs;tà pendente <lb/>in vna parte che nell'altra, ilche <lb/>non auiene in qual &longs;i voglia altro <lb/>punto della circon&longs;erenza del cer­<lb/>chin, eccettuato il D. </s> <s>Sia il centro <lb/> <arrow.to.target n="fig1"></arrow.to.target><lb/>della grauezza dell'i&longs;te&longs;&longs;o pe&longs;o, come in F, concio&longs;ia che la di&longs;ce&longs;a &longs;ia dal punto <lb/>F ver&longs;o il D, & la a&longs;ce&longs;a ver&longs;o il B, però il punto F mouera&longs;&longs;i in giù: & per­<lb/>cioche non &longs;i puote mouere al centro del mondo per linea diritta, per e&longs;&longs;er<gap/> impe­<lb/>dito dal punto C immobile per cau&longs;a della linea CF, ma ben &longs;i mouerà &longs;empre <lb/>in giù come richiede la &longs;ua natura: & e&longs;&longs;endo il D il luogo infimo, &longs;i mouerà per <lb/>la circonferenza FD finche peruenga in D, nelqual &longs;ito fermera&longs;&longs;i il pe&longs;o, & <lb/>re&longs;terà immobile, sì perche non &longs;i puote più mouere in giù per e&longs;&longs;ere attaccato al <lb/>punto C, sì anche percioche egli è &longs;o&longs;tenuto nel &longs;uo centro della grauezza. </s> <s>Et <lb/>quando F &longs;arà in D, &longs;arà &longs;imilmente la FC in DC, & in&longs;ieme à piombo <lb/>dell'orizonte. </s> <s>il pe&longs;o dunque non &longs;i fermerà giamai finche la linea CF non &longs;tia <lb/>à piombo dell'orizonte, che bi&longs;ognaua prouare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note2"></margin.target><emph type="italics"/>Per la terza pre&longs;upposta di questo.<emph.end type="italics"/></s> </p> <figure id="fig1"></figure> <p type="main"> <s>Di quì &longs;i puote cauare, che il pe&longs;o &longs;ia pur &longs;o&longs;tenuto in vn dato punto <lb/>in qual &longs;i voglia modo, non &longs;tarà fermo giamai, &longs;e non quando la <lb/>linea tirata dal centro della grauezza del pe&longs;o à quel punto, &longs;tia à <lb/>piombo dell'orizonte. </s> </p> <pb pagenum="4"/> <p type="main"> <s><emph type="italics"/>Come, po&longs;te le co&longs;e i&longs;te&longs;&longs;e, &longs;ia &longs;o&longs;tenuto <lb/>il pe&longs;o dalle linee CG CH. </s> <s>Dico <lb/>che &longs;e la tirata linea BC &longs;arà à <lb/>piombo dell'orizonte, il pe&longs;o &longs;tarà <lb/>fermo: ma &longs;e la tirata linea CF <lb/>non &longs;arà à piombo dell'orizonte, il <lb/>punto F &longs;imouerà in giù fin al D, <lb/>nel qual &longs;ito &longs;tarà fermo il pe&longs;o, <lb/>& la tirata linea CD &longs;arà à piom­<lb/>bo dell'orizonte. </s> <s>Le quali co&longs;e <lb/>tutte con laragione mede&longs;ima &longs;i pro­<lb/>uerebbono.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>PROPOSITIONE II.</s> </p> <p type="main"> <s>La bilancia egualmente di&longs;tante dall'orizonte, il cui centro &longs;tia &longs;opra <lb/>la detta bilancia, & che habbia i pe&longs;i eguali nelle &longs;tremità, & egual­<lb/>mente di&longs;tanti dal perpendicolo, &longs;e da cotale &longs;ito &longs;arà mo&longs;&longs;a, & <lb/>nell'i&longs;te&longs;&longs;o di nuouo la&longs;ciata, ritornerà, & iui re&longs;terà. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia la bilancia AB in <lb/>linea diritta egualmen <lb/>te di&longs;tante dall'orizon <lb/>te, il cui centro C &longs;ia <lb/>&longs;opra la bilancia, & <lb/>&longs;ia CD il perpendi­<lb/>colo, il quale &longs;arà à <lb/>piombo dell'orizonte: <lb/>& la di&longs;tanza DA <lb/>&longs;ia eguale alla di&longs;tan­<lb/>za DB: & &longs;iano i <lb/>pe&longs;i in AB eguali, <lb/>i centri della grauez­<lb/>za de' quali &longs;iano ne i <lb/>punti AB. </s> <s>Moua&longs;i <lb/>da que&longs;to &longs;ito la bi­<lb/>lancia AB come in EF, dapoi &longs;ia la&longs;ciata. </s> <s>Dico che la bilancia EF ritor­<lb/>neràin AB di&longs;tante egualmente dall'orizonte, & iui rimanerà. </s> <s>Hora percioche<emph.end type="italics"/> <pb/><emph type="italics"/>il punto C &longs;tà immobì <lb/>le mentre la bilancia &longs;i <lb/>moue, il punto D veni <lb/>rà à de&longs;criuere vna cir­<lb/>con&longs;erenza di cerchio, il <lb/>cui mezo diametro &longs;a­<lb/>rà CD. </s> <s>Per laqual <lb/>co&longs;a co'lcentro D, & <lb/>lo &longs;patio CD de&longs;cri­<lb/>ua&longs;i il cerchio DGH. </s> <s><lb/>Et perche CD &longs;empre <lb/>&longs;tà à piombo della bi­<lb/>lancia, mentre la bilan <lb/>cia &longs;arà in EF, la li­<lb/>nea CD &longs;arà in CG <lb/>&longs;i fattamente, che CG <lb/> <arrow.to.target n="fig2"></arrow.to.target><lb/>venga ad e&longs;&longs;ere à piombo di EF: & concio&longs;ia che AB &longs;ia diui&longs;a in due parti<emph.end type="italics"/><lb/> <arrow.to.target n="note3"></arrow.to.target> <emph type="italics"/>eguali nel punto D, & i pe&longs;iin AB &longs;iano eguali, &longs;arà etiandio il centro della <lb/>grauezza della magnitudine compo&longs;ta di que&longs;ti due corpi AB nel mezo, cioè in <lb/>D: & quando la bilancia in&longs;ieme co i pe&longs;i &longs;arà in EF, &longs;arà parimente G il cen <lb/>tro della grauezza della magnitudine compo&longs;ta di e&longs;&longs;i AB: & percioche CG <lb/>non è à piombo dell'orizonte, la grandezza compo&longs;ta de i pe&longs;i EF non rimarrà<emph.end type="italics"/><lb/> <arrow.to.target n="note4"></arrow.to.target> <emph type="italics"/>in questo &longs;ito, ma &longs;i mouerà in giù &longs;econdo il centro della grauezza &longs;ua, che è in <lb/>G, per la circonferenza GD, finche &longs;i faccia à piombo dell'orizonte, cioè finche <lb/>CG ritorni in CD. </s> <s>Et quando CG &longs;arà in CD, la linea EF (perche &longs;em­<lb/>pre &longs;tà ad angoli retti con CG) &longs;arà in AB, nelqual &longs;ito &longs;tarà &longs;erma. </s> <s>La bi-<emph.end type="italics"/><lb/> <arrow.to.target n="note5"></arrow.to.target> <emph type="italics"/>lancia dunque EF ritornerà in AB, laquale è di&longs;tante egualmente dall'orizon­<lb/>te, & iui rimarrà, che bi&longs;ognaua dimo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note3"></margin.target><emph type="italics"/>Per la quarta del primo di Archimede delle co&longs;e che pe&longs;ano egualmente.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note4"></margin.target><emph type="italics"/>Per la prima di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note5"></margin.target><emph type="italics"/>Per la prima di questo.<emph.end type="italics"/></s> </p> <figure id="fig2"></figure> <p type="head"> <s>PROPOSITIONE III.</s> </p> <p type="main"> <s>La bilancia egualmente di&longs;tante dall'orizonte, che habbia nelle &longs;tre­<lb/>mità pe&longs;i eguali, & egualmente lontani dal perpendicolo, e&longs;&longs;endo <lb/>collocato il centro di &longs;otto, rimarrà in que&longs;to &longs;ito. </s> <s>Ma &longs;e indi &longs;arà <lb/>mo&longs;&longs;a, & la&longs;ciata à ba&longs;&longs;o, &longs;i mouerà &longs;econdo la parte piu ba&longs;&longs;a. </s> </p> <pb pagenum="5"/> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia la bilancia AB in <lb/>linea diritta, egual­<lb/>mente di&longs;tante dall'ori <lb/>zonte, il cui centro C <lb/>&longs;ia di &longs;otto alla bilan­<lb/>cia, & &longs;ia CD il per­<lb/>pendicolo, ilquale &longs;arà <lb/>à piombo dell'orizon­<lb/>te, & la di&longs;tanza AD <lb/>&longs;ia eguale alla distan­<lb/>za DB, & &longs;iano in <lb/>AB pe&longs;i eguali, i cen­<lb/>tri della grauezza de' <lb/>quali &longs;iano ne' punti <lb/>AB. </s> <s>Dico primiera­<lb/>mente che la bilancia <lb/>AB &longs;tarà &longs;erma in <lb/>que&longs;to &longs;ito. </s> <s>Hor percioche AB &longs;i diuide in parti eguali nel punto D, & i <lb/>pe&longs;i po&longs;ti in AB &longs;ono eguali, &longs;egue, che il punto D &longs;ia il centro della grauez­<lb/>za della magnitudine compo&longs;ta di ambedue i corpi me&longs;&longs;i in AB; & il CD che<emph.end type="italics"/> <arrow.to.target n="note6"></arrow.to.target><lb/><emph type="italics"/>&longs;ostiene la bilancia &longs;tà à piombo dell'orizonte: Adunque la bilancia AB in <lb/>que&longs;to &longs;ito rimarrà ferma. </s> <s>Ma da que&longs;to &longs;ito moua&longs;i la bilancia AB come in <lb/>EF, & la&longs;<gap/>i&longs;i dapoi. </s> <s>Dico che la bilancia EF &longs;i mouerà dalla parte dello F. </s> <s><lb/>Et percioche il CD &longs;tà &longs;empre à piombo della bilancia, mentre la bilancia &longs;arà <lb/>in EF verrà ad e&longs;&longs;ere anche il CD in CG à piombo di EF, & il punto<emph.end type="italics"/> <arrow.to.target n="note7"></arrow.to.target><lb/><emph type="italics"/>G della magnitudine composta di EF &longs;arà il centro della grauezza, ilquale men <lb/>tre &longs;i moue de&longs;criuerà la circonferenza del cerchio DGH, il cui mezo diametro <lb/>è CD, & il centro C. </s> <s>Ma perche CG non &longs;tà à piombo dell'orizonte, la <lb/>grandezza compo&longs;ta de i pe&longs;i EF non rimarrà in questo &longs;ito, ma &longs;econdo il cen­<lb/>tro della &longs;ua grauezza &longs;i mouerà in giù per la circonferenza GH. </s> <s>La bilancia <lb/>dunque EF &longs;i mouerà in giù dalla parte dello F, che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note6"></margin.target><emph type="italics"/>Per la quar ta del primo d' Archimede delle co&longs;e che pe&longs;ano <expan abbr="egualm&etilde;te">egualmente</expan>.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note7"></margin.target><emph type="italics"/>Per la prima di questo.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE IIII.</s> </p> <p type="main"> <s>La bilancia egualmente di&longs;tante dall'orizonte, & che habbia nelle &longs;tre <lb/>mità pe&longs;i eguali, & egualmente di&longs;tanti dal centro collocato in e&longs;&longs;a <lb/>bilancia. </s> <s>Se ella indi &longs;arà mo&longs;&longs;a, ò non, douunque ella &longs;arà la&longs;cia­<lb/>ta, rimarrà. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia la bilancia nella linea <lb/>diritta AB egualmen <lb/>te di&longs;tante dall'orizon­<lb/>te, il cui centro C &longs;ia <lb/>nella i&longs;te&longs;&longs;a linea AB, <lb/>& la di&longs;tanza CA &longs;ia <lb/>eguale alla distanza <lb/>CB, & &longs;iano i pe&longs;i <lb/>AB eguali, i cui cen­<lb/>tri della grauezza &longs;tia <lb/>no ne i punti AB. </s> <s>Mo <lb/>ua&longs;i la bilancia come in <lb/>DE, & iui &longs;ia la&longs;cia­<lb/>ta. </s> <s>Dico primamen­<lb/> <arrow.to.target n="fig3"></arrow.to.target><lb/>te che la bilancia DE non &longs;i mouerà, & rimarrà in quel &longs;ito. </s> <s>Hor percioche i <lb/>pe&longs;i AB &longs;ono eguali, &longs;arà il centro della grauezza della magnitudine compo&longs;ta <lb/>delli due pe&longs;i A & B in C. </s> <s>Per laqual co&longs;a l'i&longs;te&longs;&longs;o punto C &longs;arà il centro <lb/>della bilancia, & il centro della grauezza di tutto il pe&longs;o. </s> <s>Et percioche il centro <lb/>della bilancia che è C, mentre la bilancia AB in&longs;ieme co'pe&longs;i &longs;i moue in DE, <lb/>rimane immobile, non &longs;i mouerà ne anche il centro della grauezza, che è l'i&longs;te&longs;&longs;o C. </s> <s><lb/>Adunque ne anche la bilancia DE &longs;i mouerà per la diffinitione del centro della <lb/>grauezza, e&longs;&longs;endo in e&longs;&longs;o appiccata. </s> <s>L'i&longs;te&longs;&longs;o accade parimente &longs;tando la bilancia <lb/>AB egualmente di&longs;tante dall'orizonte, ouero e&longs;&longs;endo in qual &longs;i voglia altro &longs;ito. </s> <s><lb/>Rimarrà dunque la bilancia oue &longs;arà la&longs;ciata, che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig3"></figure> <p type="main"> <s><emph type="italics"/>Benche habbiamo con&longs;iderato nelle co&longs;e predette le grauezze &longs;olamente delle magni­<lb/>tudini, le quali &longs;ono po&longs;te nelle &longs;tremità della bilancia, &longs;enza la grauezza della bi­<lb/>lancia; niente di manco per e&longs;&longs;ere anche le braccia della bilancia eguali, auenir à lo <lb/>i&longs;te&longs;&longs;o alla bilancia, con&longs;iderata la &longs;ua grauezza in&longs;ieme co' pe&longs;i, ouero &longs;enza pe&longs;i, <lb/>percioche il centro iste&longs;&longs;o della grauezza &longs;enza pe&longs;i &longs;arà anco centro della grauez­<lb/>za della bilancia &longs;ola. </s> <s>Similmente &longs;e li pe&longs;i &longs;aranno appiccati nelle &longs;tremità del­<lb/>la bilancia, come &longs;uole far&longs;t, aùerrà l'iste&longs;&longs;o, purche le linee tirate da i punti oue &longs;o­<lb/>no attaccati i pe&longs;i ver&longs;o i centri delle grauezze, (moua&longs;i la bilancia in qual &longs;i vo­<lb/>gliamodo) vadano à concorrere nel centro del mondo, peroche doue &longs;ono attaccati <lb/>i pe&longs;i in questa maniera, iui grauano, come &longs;e in quegli &longs;te&longs;&longs;i punti baue&longs;&longs;ero i cen <lb/>tri delle grauezze. </s> <s>Oltre à ciò poßiamo con&longs;iderare le co&longs;e che &longs;eguono in tut­<lb/>to al modo i&longs;te&longs;&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s> <arrow.to.target n="note8"></arrow.to.target><emph type="italics"/>Ma percioche à que&longs;ta vltima conchiu&longs;ione molte co&longs;e dette da alcuni, che &longs;entono al­<lb/>tramente, paiono contra&longs;tare; però in cote&longs;ta parte egli &longs;arà bi&longs;ogno dimorare <lb/>alquanto, & &longs;econdo le mie forze non &longs;olo farò opra di difendere la propria <lb/>mia &longs;entenza, ma Archimede ancora, ilquale &longs;embra e&longs;&longs;ere &longs;tato in que&longs;to i&longs;te&longs;­<lb/>&longs;o parere.<emph.end type="italics"/></s> </p> <pb pagenum="6"/> <p type="margin"> <s><margin.target id="note8"></margin.target><emph type="italics"/>Giord. </s> <s>de' pe&longs;i. </s> <s>Il Car dano della &longs;ottigliezza. </s> <s>Il Tartaglia de' que&longs;iti, & <expan abbr="inu&etilde;tioni">inuentioni</expan><emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;te le co&longs;e i&longs;te&longs;&longs;e, &longs;ia <lb/>tirata la linea FCG <lb/>à piombo di AB, & <lb/>dell'orizonte: & col <lb/>centro C, & lo &longs;pa­<lb/>tio CA &longs;ia de&longs;crit­<lb/>to il cerchio ADFB <lb/>EG: &longs;aranno i punti <lb/>ADBE nella circon <lb/>ferenza del cerchio, <lb/>per e&longs;&longs;ere le braccia <lb/>della bilancia egual</s> <s>/>& percioche conuen­<lb/>gono que&longs;ti autori in <lb/>vna &longs;entenza, affer­<lb/>mando, che la bilan­<lb/>cia DE non &longs;i moue <lb/>in FG, ne rimane in<emph.end type="italics"/><lb/> <arrow.to.target n="fig4"></arrow.to.target><lb/><emph type="italics"/>DE, maritornanellalinea AB egualmente di&longs;tante dall'orizonte, mo&longs;trerò que <lb/>&longs;ta loro opinione non potere à modo alcuno &longs;tare. </s> <s>Percioche &longs;e egli è vero quel <lb/>che dicono, ouero auenir à questo effetto per e&longs;&longs;ere il pe&longs;o D più graue del pe&longs;o E, <lb/>ouero &longs;e li pe&longs;i &longs;ono eguali, le di&longs;tanze nelle quali &longs;ono po&longs;ti, non &longs;aranno eguali, <lb/>cioè la CD non &longs;arà eguale alla CE, ma più grande. </s> <s>Ma che i pe&longs;i col­<lb/>locati in DE &longs;iano eguali, & la di&longs;tanza CD &longs;ia eguale alla di&longs;tanza CE, è <lb/>chiaro dalla pre&longs;uppo&longs;ta. </s> <s>Hor perche dicono che il pe&longs;o po&longs;to in D in quel &longs;i­<lb/>to è più graue del pe&longs;o po&longs;to in E nell altro &longs;ito da ba&longs;&longs;o: mentre i pe&longs;i &longs;ono in <lb/>DE, non &longs;arà il punto C piu centro della grauezza, imperoche non stanno fer­<lb/>mi &longs;e &longs;ono attaccati al C, ma &longs;arà nella linea CD per la terza del primo di At <lb/>chimede delle co&longs;e che pe&longs;ano egualmente. </s> <s>Non &longs;arà già nella CE per e&longs;&longs;ere il <lb/>pe&longs;o D più graue del pe&longs;o E: &longs;ia dunque in H, nelquale &longs;e &longs;aranno attacca­<lb/>ti, rimarranno. </s> <s>Et percioche il centro della grauezza de' pe&longs;i congiunti in AB <lb/>&longs;tà nel punto C: ma de' pe&longs;i po&longs;tiin DE il punto è H: mentre dunque i pe&longs;i <lb/>AB &longs;i muouono in DE, il centro della grauezza C moueraßi ver&longs;o D, & <lb/>s'appre&longs;&longs;er à più da vicino al D, ilche è impoßibile, per mantenere i pe&longs;i vname­<lb/>de&longs;ima di&longs;tanza fra loro: peroche il centro della grauezza di cia&longs;cun corpo &longs;tà &longs;em­<lb/>pre nel mede&longs;imo &longs;ito per ri&longs;petto al &longs;uo corpo. </s> <s>Et quantunque il punto C &longs;ia il<emph.end type="italics"/> <arrow.to.target n="note9"></arrow.to.target><lb/><emph type="italics"/>centro della grauezza di due corpi A. & B, tuttauia per e&longs;&longs;ere mediante la bi­<lb/>lancia co&longs;i giunti in&longs;ieme, che &longs;empre &longs;i trouano nell'iste&longs;&longs;o modo; però il punto C<emph.end type="italics"/> <arrow.to.target n="note10"></arrow.to.target><lb/><emph type="italics"/>&longs;arà co&longs;i centro della grauezza loro, come &longs;e fo&longs;&longs;e vna &longs;ola magnitudine; percio­<lb/>che la bilancia in&longs;ieme co' pe&longs;i fa vn &longs;olo corpo continuo, il cui centro della grauez <lb/>za &longs;empre &longs;tarà nel mezo. </s> <s>Non è dunque il pe&longs;o po&longs;to in D più graue del pe­<lb/>&longs;o po&longs;to in E. </s> <s>Che &longs;e dice&longs;&longs;ero il centro della grauezza non nella linea CD, ma<emph.end type="italics"/> <pb/><emph type="italics"/>nella CE douer e&longs;&longs;ere, auerrà l'i&longs;te&longs;&longs;o &longs;allo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note9"></margin.target><emph type="italics"/>Per la &longs;ecom da &longs;upposta di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note10"></margin.target><emph type="italics"/>Per la quar ta del primo di Archime de delle co&longs;e che pe&longs;ano egualmente.<emph.end type="italics"/></s> </p> <figure id="fig4"></figure> <p type="main"> <s><emph type="italics"/>Di più &longs;e il pe&longs;o D &longs;i mouer à in giù, mouer à il pe&longs;o E in sù. </s> <s>Adunque vn pe&longs;o <lb/>più graue di E nel mede&longs;imo &longs;ito pe&longs;erà tanto quanto il pe&longs;o D, & auerr à che <lb/>co&longs;e graui di&longs;uguali, po&longs;te in eguale distanza pe&longs;eranno egualmente. </s> <s>Aggiun­<lb/>ga&longs;i dunque al pe&longs;o E qualche co&longs;a graue, &longs;i &longs;attamente, che contrape&longs;i al D &longs;e <lb/>nel C &longs;aranno attac <lb/>cati. </s> <s>Ma e&longs;&longs;endo &longs;ta­<lb/>to di &longs;opra mo&longs;trato <lb/>il punto C e&longs;&longs;ere il cè-<emph.end type="italics"/><lb/> <arrow.to.target n="note11"></arrow.to.target> <emph type="italics"/>tro della grauezza di <lb/>pe&longs;i eguali po&longs;ti in <lb/>DE; &longs;e dunque il pe­<lb/>&longs;o. </s> <s>E &longs;arà più graue <lb/>del pe&longs;o D, &longs;arà anche <lb/>il centro della grauez <lb/>za nella linea C</s> <s>/>& &longs;ia que&longs;to centro <lb/>il<emph.end type="italics"/> K. <emph type="italics"/>Ma per la diffi­<lb/>nitione del centro del <lb/>la grauezza, &longs;e li pe&longs;i <lb/>&longs;aranno appiccati al<emph.end type="italics"/><lb/>K, <emph type="italics"/>staranno fermi. </s> <s><lb/>Dunque &longs;e &longs;aranno<emph.end type="italics"/><lb/> <arrow.to.target n="fig5"></arrow.to.target><lb/><emph type="italics"/>appiccati al C, non &longs;taranno fermi, che è contra la pre&longs;uppo&longs;ta: ma il pe&longs;o E &longs;i<emph.end type="italics"/><lb/> <arrow.to.target n="note12"></arrow.to.target> <emph type="italics"/>mouer à in giù. </s> <s>Che &longs;e appiccati al C pe&longs;a&longs;&longs;ero ancora egualmente, na&longs;cerebbe <lb/>che di vna magnitudine, due &longs;arebbono i centri della grauezza, che è impo&longs;&longs;ibile. </s> <s><lb/>Adunque il pe&longs;o po&longs;to in E più graue di quello che è in D, non pe&longs;er à tanto <lb/>quanto il D attaccando&longs;i al punto C. </s> <s>I pe&longs;i dunque eguali po&longs;ti in DE, attac­<lb/>cati nel centro della loro grauezza pe&longs;eranno egualmente, & &longs;taranno immobili, <lb/>che &longs;u proposto di mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note11"></margin.target><emph type="italics"/>Per laterza del primo di Archimede delle co&longs;e che pe&longs;ano egual mente.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note12"></margin.target><emph type="italics"/>Per la prima &longs;upposta di questo.<emph.end type="italics"/></s> </p> <figure id="fig5"></figure> <p type="main"> <s> <arrow.to.target n="note13"></arrow.to.target> <emph type="italics"/>A que&longs;ta vltima &longs;conueneuolezza ri&longs;pondono, dicendo e&longs;&longs;ere impo&longs;&longs;ibile aggiungere al <lb/>lo E &longs;i picciolo pe&longs;o, che in ogni modo &longs;e ben &longs;i appiccano al C, il pe&longs;o E non <lb/>&longs;i moua &longs;empre in giù ver&longs;o il G. </s> <s>La qual co&longs;a habbiamo noi pre&longs;uppo&longs;to poter&longs;i <lb/>fare, & credeuamo poter&longs;i fare: Peroche quel che è di più del pe&longs;o D &longs;opra <lb/>il pe&longs;o E, hauendo ragione, & parte di quantità, &longs;i imaginauamo non &longs;olamente <lb/>e&longs;&longs;ere minimo, ma ancora poter&longs;i diuidere in infinito, il che eßi per certo non &longs;ola­<lb/>mente minimo, ma ne anche e&longs;&longs;ere minimo, non potendo&longs;i ritrouare, &longs;i s&longs;orzano di <lb/>mo&longs;tr are in que&longs;ta maniera.<emph.end type="italics"/></s> </p> <pb pagenum="7"/> <p type="margin"> <s><margin.target id="note13"></margin.target><emph type="italics"/>Il Tartaglia nella &longs;esta propo&longs;itione del quate li bre.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Pongan&longs;i le co&longs;e iste&longs;&longs;e <lb/>& da<gap/>i punti DE <lb/>&longs;iano tirate le linee <lb/>DHE<emph.end type="italics"/>K <emph type="italics"/>à piombo <lb/>dell'orizonte, & &longs;ia <lb/>vn'altro cerchio L <lb/>DM, il cui centro <lb/>&longs;ia N, ilquale toc<emph.end type="italics"/> <arrow.to.target n="note14"></arrow.to.target><lb/><emph type="italics"/>chi FDG nel pun <lb/>to D, & &longs;ia eguale<emph.end type="italics"/> <arrow.to.target n="note15"></arrow.to.target><lb/><emph type="italics"/>ad FDG. </s> <s>Sarà <lb/>NC linea retta: & <lb/>perche l'angolo<emph.end type="italics"/> K <lb/><emph type="italics"/>EC è eguale all'an­<lb/>golo HDN, & <lb/>l'angolo CEG è pa <lb/>rimente eguale al­<lb/>l'angolo NDM,<emph.end type="italics"/><lb/> <arrow.to.target n="fig6"></arrow.to.target><lb/><emph type="italics"/>peroche egli è contenuto da mezi diametri, & da circonferenze eguali: &longs;arà il re­<lb/>stante angolo & mi&longs;to<emph.end type="italics"/> K<emph type="italics"/>EG eguale al re&longs;tante angolo & mi&longs;to HDM. </s> <s>Et per­<lb/>cioche pre&longs;uppongono, che quanto è minore l'angolo contenuto dalla linea tirata à <lb/>piombo dell'orizonte, & dalla circonferenza, tanto in quel &longs;ito e&longs;&longs;ere anco più gra <lb/>ue il pe&longs;o. </s> <s>Talche &longs;i come l'angolo contenuto da HD, & dalla circonferenza <lb/>DG, è minore dell'angolo<emph.end type="italics"/> K<emph type="italics"/>EG, cioè dell'angolo HDM, co&longs;i &longs;econdo que&longs;ta <lb/>proportione il pe&longs;o po&longs;to in D &longs;ia più graue di quello che &longs;tà in E. </s> <s>Mala pro­<lb/>portione dell'angolo MHD all'angolo HDG è minore di qual &longs;i voglia altra <lb/>proportione, che &longs;i troui tra la maggiore, & minore quantità: Adunque la pro­<lb/>portione de i pe&longs;i DE &longs;arà la minima di tutte le proportioni, anzinon &longs;arà qua&longs;i <lb/>ne anche proportione, e&longs;&longs;endo la minima di tutte le proportioni. </s> <s>Che la propor­<lb/>tione di MDH ver&longs;o HDG &longs;ia di tutte la minima, mo&longs;trano con que&longs;ta ne­<lb/>ce&longs;&longs;aria ragione, peroche MHD &longs;upera HDG con angolo di linea curua, che <lb/>è MGD, ilquale angolo è il minimo di tutti gli angoli fatti di linee rette: ne po­<lb/>tendo&longs;i dare angolo minore di MGD &longs;arà la proportione di MDH ver&longs;o HDG <lb/>la minima di tutte le proportioni. </s> <s>Laqual ragione pare e&longs;&longs;ere grandemente friuo­<lb/>la, peroche quantunque l'angolo MDG &longs;ia di tutti gli angoli fatti di linee rette <lb/>il minore, non perciò &longs;egue totalmente egli e&longs;&longs;ere di tutti gli angoli il minimo, im-<emph.end type="italics"/> <arrow.to.target n="note16"></arrow.to.target><lb/><emph type="italics"/>peroche &longs;ia dal punto D tirata la linea DO à piombo di NC, ambedue que­<lb/>ste toccberanno le circonferenze LDMFDG nel punto D. </s> <s>Ma percioche le <lb/>circonferenze &longs;ono eguali, &longs;arà l'angolo MDO misto eguale all'angolo ODG mi­<lb/>&longs;to. </s> <s>L'vno de gli angoli dunque, cioè ODG &longs;arà minore di MDG, cioè minore<emph.end type="italics"/> <arrow.to.target n="note17"></arrow.to.target><lb/><emph type="italics"/>del minimo. </s> <s>Dapoi l'angolo ODH &longs;arà minore dell'angolo MDH. </s> <s>Per laqual co&longs;a <lb/>ODH haurà proportione minore all'angolo HDG, che MDH all'i&longs;te&longs;&longs;o<emph.end type="italics"/> <pb/><emph type="italics"/>HDG. </s> <s>Dara&longs;&longs;i dunque la proportione anco minore della minima, laquale mostre­<lb/>remo dauantaggio in in&longs;inito minore in questo modo. </s> <s>De&longs;criua&longs;i il cerchio DR, <lb/>il cui centro &longs;ia E, & il mezo diametro ED, la circonferentia DR tocche-<emph.end type="italics"/><lb/> <arrow.to.target n="note18"></arrow.to.target> <emph type="italics"/>rà la circonferenza <lb/>DG nel punto D, <lb/>& la linea DO nel<emph.end type="italics"/><lb/> <arrow.to.target n="note19"></arrow.to.target> <emph type="italics"/>punto D. </s> <s>Per laqual <lb/>co&longs;a minore &longs;arà l'an <lb/>golo RDG dell'an­<lb/>golo ODG, & &longs;i­<lb/>milmente l'angolo R <lb/>DH dell'angolo O <lb/>DH. </s> <s>Adunque ha­<lb/>uer à minore propor­<lb/>tione RDH ad HD <lb/>G di quel che haurà <lb/>ODH ad HDG. </s> <s><lb/>Pigli&longs;i dapoi tra E <lb/>& C, come &longs;i vuo­<lb/>le, il punto P, dal <lb/>quale nella di&longs;tanza<emph.end type="italics"/><lb/> <arrow.to.target n="fig7"></arrow.to.target><lb/><emph type="italics"/>di PD &longs;i de&longs;criua vn'altra circonferenza DQ, laquale toccherà la circonferen­<lb/>tia DR, & la circonferentia DG nel punto D, & l'angolo QDH &longs;arà mi <lb/>nore dell'angolo RDH. </s> <s>Adunque QDH haurà proportione minore ad HDG <lb/>che RDH ad HDG, & nell'i&longs;te&longs;&longs;o modo in tutto, &longs;e tra il C & il P &longs;i tor­<lb/>rà vn'altro punto, & tra que&longs;to, & il C vn'altro, & co&longs;i &longs;ucceßiuamente &longs;i de­<lb/>&longs;criueranno infinite circonferentie tra DO, & la circonferenza DG: dalle quali <lb/>troueremo &longs;empre la proportione minore in infinito: & co&longs;i &longs;egue, che la propor­<lb/>tione del pe&longs;o po&longs;to in D al pe&longs;o po&longs;to in E non &longs;ia tanto picciola, che non &longs;i <lb/>po&longs;&longs;a ritrouarla &longs;empre minore in infinito. </s> <s>Et perche l'angolo MDG &longs;i puote <lb/>diuidere in infinito, &longs;i potrà anche diuidere quel più di grauezza che ha il D &longs;o­<lb/>pra lo E in infinito.<emph.end type="italics"/></s> </p> <pb pagenum="8"/> <p type="margin"> <s><margin.target id="note14"></margin.target><emph type="italics"/>Per la &longs;ecen da del terzo<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note15"></margin.target><emph type="italics"/>Per la vige &longs;imanona del prim<gap/>.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note16"></margin.target><emph type="italics"/>Per la decima ottaua del terzo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note17"></margin.target><emph type="italics"/>Per la ottaua del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note18"></margin.target><emph type="italics"/>Per la vnde cima del ter zo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note19"></margin.target><emph type="italics"/>Per la decima ottaua del terzo.<emph.end type="italics"/></s> </p> <figure id="fig6"></figure> <figure id="fig7"></figure> <p type="main"> <s><emph type="italics"/>Ne bi&longs;ogna trala&longs;ciare, che <lb/>eglino hanno pre&longs;upp o&longs;to <lb/>nella demo&longs;tratione l'ango <lb/>lo<emph.end type="italics"/> K<emph type="italics"/>EG e&longs;&longs;er maggiore del <lb/>l'angolo HDC, come co <lb/>&longs;a nota: il che ben è vero &longs;e <lb/>DHE<emph.end type="italics"/>K <emph type="italics"/>&longs;ono fra loro e­<lb/>gualmente di&longs;tanti. </s> <s>Ma <lb/>percioche, come eßi pari­<lb/>mente pre&longs;uppongono, le <lb/>linee DHE<emph.end type="italics"/>K <emph type="italics"/>&longs;i vanno à <lb/>trouare nel centro del mon <lb/>do, le linee DHE<emph.end type="italics"/>K <emph type="italics"/>non <lb/>&longs;aranno egualmente di&longs;tan <lb/>ti giamai, et <expan abbr="l'ãgolo">l'angolo</expan><emph.end type="italics"/> K<emph type="italics"/>EG <lb/>non &longs;olo non &longs;arà maggio­<lb/>re dall'angolo HDG, ma <lb/>minore. </s> <s>Come per gra­<lb/>tia di e&longs;&longs;empio, &longs;ia tirata la <lb/>linea FG &longs;in al centro del <lb/>mondo, che &longs;ia S, & con <lb/>giungan&longs;i DS ES. </s> <s>Egli <lb/>è da mostrare l'angolo SE <lb/>G e&longs;&longs;ere minore dell'ango <lb/>lo SDG. </s> <s>Tiri&longs;i dal punto <lb/>E la linea ET, che toc­<lb/>chi il cerchio DGEF, & <lb/>dall'i&longs;te&longs;&longs;o punto &longs;ia tirata <lb/>la EV egualmente di&longs;tan<emph.end type="italics"/><lb/> <arrow.to.target n="fig8"></arrow.to.target><lb/><emph type="italics"/>te da DS: Percioche dunque EVDS &longs;ono traloro egualmente di&longs;tanti, &longs;imil­<lb/>mente ET DO &longs;ono egualmente di&longs;tanti: &longs;arà l'angolo VET eguale all'ango­<lb/>lo SDO: & l'angolo TEG eguale all'angolo ODM, per e&longs;&longs;ere contenuto da <lb/>linee toccanti la circonferenza, & da circonferenze eguali. </s> <s>Tutto l'angolo dun­<lb/>que VEG &longs;arà eguale all'angolo SDM. </s> <s>Leui&longs;i via dall'angolo SDM l'ango <lb/>lo di linee curue MDG: & dall'angolo VEG leui&longs;i via l'angolo VES, & <lb/>l'angolo VES fatto di linee rette è maggiore dell'angolo MDG fatto di linee <lb/>curue; &longs;arà il re&longs;tante angolo SEG minore dell'angolo SDG. </s> <s>Per laqual co&longs;a <lb/>dalle pre&longs;uppo&longs;te loro non &longs;olo il pe&longs;o posto in D &longs;arà più graue del pe&longs;o po&longs;to <lb/>in E, ma per lo contrario il pe&longs;o E &longs;arà più graue dell'i&longs;te&longs;&longs;o D.<emph.end type="italics"/></s> </p> <pb/> <figure id="fig8"></figure> <p type="main"> <s><emph type="italics"/>Producono tutta via <lb/>ragioni con le quali <lb/>&longs;i sforzano di mo­<lb/>&longs;trare, che la bilan­<lb/>cia DE ritorna per <lb/>neceßità in AB e­<lb/>gualmente distante <lb/>dall'orizonte. </s> <s>Pri­<lb/>ma dimo&longs;trano l'i­<lb/>&longs;te&longs;&longs;o pe&longs;o e&longs;&longs;ere più <lb/>graue in A, che <lb/>in altro &longs;ito, che <lb/>chiamano &longs;ito della <lb/>egualità, e&longs;&longs;endo la <lb/>linea AB egual­<lb/>mente di&longs;tante dal­<lb/>l'orizonte. </s> <s>Da­<lb/>poi quanto è più da<emph.end type="italics"/><lb/> <arrow.to.target n="fig9"></arrow.to.target><lb/><emph type="italics"/>pre&longs;&longs;o allo A, tanto e&longs;&longs;ere piu graue di qual &longs;i voglia altro più da lontano, cioè <lb/>il pe&longs;o po&longs;to in A e&longs;&longs;ere più graue, che in D; & in D, che in L: & &longs;imil­<lb/>mente in A più graue, che in N; & in N più graue, che in M. </s> <s>Con&longs;ide-<emph.end type="italics"/><lb/> <arrow.to.target n="note20"></arrow.to.target> <emph type="italics"/>rando &longs;olamente vn pe&longs;o in vno delle braccia in sù, ouero in giù mo&longs;&longs;o. </s> <s>Percio­<lb/>che dicono, po&longs;ta la trutina della bilancia in CF, il pe&longs;o me&longs;&longs;o in A è più lunge <lb/>dalla trutina che in D; & in D più lunge, che in L: peroche tirate le linee DO<emph.end type="italics"/><lb/> <arrow.to.target n="note21"></arrow.to.target> <emph type="italics"/>LP à piombo di CF, la linea AC re&longs;ta maggiore di DO, & DO di e&longs;&longs;a LP, <lb/>& auiene l'i&longs;te&longs;&longs;o ne i punti NM. </s> <s>Dapoi dicono da qual luogo il pe&longs;o &longs;i mo­<lb/>ue più velocemente, iui è più graue: ma egli &longs;i moue più velocemente dallo<emph.end type="italics"/><lb/> <arrow.to.target n="note22"></arrow.to.target> <emph type="italics"/>A, che da altro &longs;ito; adunque egli è più graue nello A. </s> <s>Con &longs;imile mo­<lb/>do, quanto più egli è da pre&longs;&longs;o allo A, tanto più velocemente &longs;i moue: <lb/>adunque nel D &longs;arà più graue, che in L. </s> <s>L'altra cagione poi che cauano dal mo-<emph.end type="italics"/><lb/> <arrow.to.target n="note23"></arrow.to.target> <emph type="italics"/>uimento più diritto, & più torto è, che quanto il pe&longs;o di&longs;cende più diritto in archi <lb/>eguali, pare e&longs;&longs;er anco più graue; concio&longs;ia che il pe&longs;o e&longs;&longs;endo libero, & &longs;ciolto, &longs;i <lb/>moua di &longs;ua propria natura per lo diritto; ma in A egli di&longs;cende più dirittamen<emph.end type="italics"/><lb/> <arrow.to.target n="note24"></arrow.to.target> <emph type="italics"/>te; dunque in A &longs;arà più graue, & dimo&longs;trano ciò pigliando l'arco AN egua­<lb/>le all'arco LD. & da i punti NL &longs;iano tirate le linee NRLQ egualmente di­<lb/>&longs;tanti dalla linea FG, laquale chiamano anche della direttione; & quelle altre &longs;e­<lb/>gheranno le linee ABDO in QR, & dal punto N &longs;ia tirata la NT à piombo <lb/>di FG: Dimo&longs;trano veramente LQ e&longs;&longs;ere eguale à PO, & NR ad e&longs;&longs;a CT, <lb/>& la linea NR e&longs;&longs;er maggiore di <expan abbr="Lq.">Lque</expan> Hor percioche la di&longs;ce&longs;a del pe&longs;o dallo A <lb/>fin ad N per la circonferentia di AN trapa&longs;&longs;a maggior parte della linea FG, <lb/>(che eßi chiamano pigliare di diritto) che la di&longs;ce&longs;a di L in D per la circonferenza <lb/>LD; concio&longs;ia che la di&longs;ce&longs;a AN trapaßi la linea CT, ma la di&longs;ce&longs;a LD la linea<emph.end type="italics"/> <pb pagenum="9"/><emph type="italics"/>PO, & CT è maggiore di PO, la di&longs;ce&longs;a di AN &longs;arà più diritta, che la di­<lb/>&longs;ce&longs;a di LD: &longs;arà dunque più graue il pe&longs;o po&longs;to in A, che in L, ouero in qual <lb/>&longs;i voglia altro &longs;ito, & nell'i&longs;te&longs;&longs;o modo dimo&longs;trano, che quanto il pe&longs;o è più vicino <lb/>allo A, è più graue; cioè &longs;iano le circonferenze LD DA traloro eguali, & <lb/>dal punto D &longs;ia tirata la linea DR à piombo di AB; &longs;arà la DR eguale al-<emph.end type="italics"/> <arrow.to.target n="note25"></arrow.to.target><lb/><emph type="italics"/>la CO. </s> <s>& dimo­<lb/>&longs;trano po&longs;cia, che <lb/>la linea DR è mag <lb/>giore della LQ, & <lb/>dicono che la &longs;ce&longs;a <lb/>di DA prende più <lb/>di &longs;ce&longs;a diritta, che <lb/>non fa LD, pe­<lb/>roche è maggiore <lb/>la linea CO, che <lb/>la OT<gap/> Per la­<lb/>qual co&longs;a<gap/> pe&longs;o &longs;a <lb/>rà più graue in D, <lb/>che in L, ilche pa <lb/>rimente auiene ne <lb/>punti NM. </s> <s>& <lb/>co&longs;i il pre&longs;uppo&longs;to, <lb/>per loquale dimo-<emph.end type="italics"/><lb/> <arrow.to.target n="fig10"></arrow.to.target><lb/><emph type="italics"/>&longs;trano la bilancia DE ritornare in AB a&longs;fermano come noto, & manife&longs;to; cioè<emph.end type="italics"/> <arrow.to.target n="note26"></arrow.to.target><lb/><emph type="italics"/>che &longs;econdo il &longs;ito il pe&longs;o è tanto più graue, quanto nel mede&longs;imo &longs;ito manco tor­<lb/>ta è la &longs;ce&longs;a: & la cagione di cotal ritorno dicono e&longs;&longs;ere que&longs;ta; peroche la &longs;ce&longs;a del <lb/>pe&longs;o po&longs;to in D è più diritta della &longs;ce&longs;a del pe&longs;o po&longs;to in E, per pigliare il pe&longs;o<emph.end type="italics"/> <arrow.to.target n="note27"></arrow.to.target><lb/><emph type="italics"/>di E manco della direttione in de&longs;cendendo che non fa il pe&longs;o di D pur nel di&longs;cen <lb/>dere: Come &longs;e l'arco EV &longs;ia eguale à DA, & &longs;iano tirate VHET à piom <lb/>bo di FG; &longs;arà maggiore DR di TH. </s> <s>Per laqual co&longs;a per la pre&longs;uppo&longs;ta il pe<emph.end type="italics"/> <arrow.to.target n="note28"></arrow.to.target><lb/><emph type="italics"/>&longs;o me&longs;&longs;o in D per ri&longs;petto al &longs;ito &longs;arà più graue del pe&longs;o me&longs;&longs;o in E. </s> <s>Adunque <lb/>il pe&longs;o me&longs;&longs;o in D e&longs;&longs;endo più graue &longs;i mouerà in giù, & il pe&longs;o po&longs;to in E in <lb/>&longs;u fin che la bilancia DE ritorni in AB.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note20"></margin.target><emph type="italics"/>Il Cardano nel primo della &longs;ottigliezza.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note21"></margin.target><emph type="italics"/>Giordano nella quarta propo&longs;itione<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note22"></margin.target><emph type="italics"/>Il Tartaglia nella quinta propo&longs;itione.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note23"></margin.target><emph type="italics"/>Il Cardano. </s> <s>Giordano al la propo&longs;itio ne quarta.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note24"></margin.target><emph type="italics"/>Il Tartaglia alla pro po&longs;itione<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note25"></margin.target><emph type="italics"/>Per la trige &longs;imaquarta del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note26"></margin.target><emph type="italics"/>Giordane nella quarta pre &longs;apoosta<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note27"></margin.target><emph type="italics"/>Giordano nella &longs;econda propo&longs;iti<gap/> ne.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note28"></margin.target><emph type="italics"/>Il Tartaglia nella quinta propo&longs;icione.<emph.end type="italics"/></s> </p> <figure id="fig9"></figure> <figure id="fig10"></figure> <p type="main"> <s><emph type="italics"/>L'altra ragione ancora di que&longs;to ritorno è, che <expan abbr="quãdo">quando</expan> la trutina della bilancia è &longs;opra<emph.end type="italics"/> <arrow.to.target n="note29"></arrow.to.target><lb/><emph type="italics"/>dilei in CF; la linea CG è la meta: & percio che l'angolo GCD è maggiore <lb/>dell'angolo GCE, & l'angolo maggiore dalla meta rende più graue il pe&longs;o: adun­<lb/>que &longs;tando la trutina della bilancia di &longs;opra &longs;arà più graue il pe&longs;o in D, che in E, <lb/>& perciò il D ritorner à nello A, & lo E nel B.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note29"></margin.target><emph type="italics"/>Il Cardano.<emph.end type="italics"/></s> </p> <p type="main"> <s>Meta è pur voce Latina co&longs;tumata da gli antichi ne i giuo chi, & conte&longs;e fatte ne i cer <lb/>chi murati, & ne i Theatri, percio che il principio, oue &longs;i dauano le mo&longs;&longs;e a' corri­<lb/>tori, &longs;i chiamaua Carcere, & il fine Meta; di modo, che meta viene à dire termine <lb/>& fine: & piu in altro &longs;ignificato il luogo piu ba&longs;&longs;o, & in&longs;imo. </s> <s>Hor qui &longs;i puote <pb/>intendere ad ambidue i modi, cioè che la linea CG &longs;ia la meta, cioè il termine <lb/>& fine, nelquale ha da peruenire il pe&longs;o collo cato nella bilancia; ouero il luogo <lb/>infimo della circonferenza, alquale capita il pe&longs;o per natura. </s> <s>Doue &longs;criue l'Auto <lb/>re l'angolo maggiore dalla Meta, vuol dire l'angolo, che fa il braccio della bilan­<lb/>cia con la Meta CG. </s> </p> <p type="main"> <s><emph type="italics"/>Et co&longs;i <expan abbr="cõ">com</expan> que&longs;te ragioni &longs;i sforzano dimo&longs;trare la bilancia DE ritornare in AB; le <lb/>quali al parer mio &longs;i po&longs;&longs;ono ageuolmente &longs;oluere.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Primieramente dunque in quanto s'appartiene alle ragioni, che dicono il pe&longs;o me&longs;&longs;o <lb/>in A e&longs;&longs;ere piu graue, che in altro &longs;ito, lequali cauano dalla di&longs;tanza piu da lonta­<lb/>no, & piu da pre&longs;&longs;o della linea FG, & dal mouimento piu veloce, & piu diritto <lb/>dal punto A. </s> <s>In prima non dimo&longs;trano veramente perche il pe&longs;o &longs;i moua piu velo­<lb/>cemente dallo A, che da al tro &longs;ito. </s> <s>ne perche &longs;ia maggiore CA di DO, & DO <lb/>di LP, per que&longs;to, come per vera cagione, &longs;egue il pe&longs;o po&longs;to in A e&longs;&longs;ere piu gra­<lb/>ue di quello, che è in D, & quello di D, di quel che &longs;tà in L, percioche non &longs;i queta <lb/>l'intelletto, &longs;e di ciò altra cagione non &longs;i dimo&longs;tra, parendo &longs;egno piu to&longs;to, che vera <lb/>cagione. </s> <s>Quello ste&longs;&longs;o accade parimente all'altra ragione, laquale adducono dal <lb/>mouimento piu diritto, & piu torto. </s> <s>Oltre à ciò tutte quelle co&longs;e, che per&longs;uadono <lb/>per via del <expan abbr="mouim&etilde;">mouimen</expan> <lb/>to piu veloce, & <lb/>piu tardo il pe&longs;o in <lb/>A e&longs;&longs;ere piu graue, <lb/>che in D, non per­<lb/>ciò dimo &longs;trano, che <lb/>il pe&longs;o in A, in <expan abbr="quã">quam</expan> <lb/>to è in A, &longs;ia piu <lb/>graue del pe&longs;o D, in <lb/>quanto è in D, ma <lb/>in quanto &longs;i parte <lb/>da i punti D A.</s> <s>Onde, <expan abbr="auãti">auanti</expan> che piu <lb/>oltre &longs;i proceda, pri <lb/>ma dimo&longs;trerò, che <lb/>il pe&longs;o quanto egli <lb/>è piu da pre&longs;&longs;o ad <lb/>FG manco graua, <lb/>&longs;i in quanto egli &longs;tà <lb/>nel &longs;ito, oue &longs;i ritro<emph.end type="italics"/><lb/> <arrow.to.target n="fig11"></arrow.to.target><lb/><emph type="italics"/>ua, come anche in quanto &longs;i parte da quello: & in&longs;ieme, che egli è fal&longs;o il pe&longs;o e&longs;&longs;ere <lb/>piu graue in A, che in altro &longs;ito.<emph.end type="italics"/></s> </p> <figure id="fig11"></figure> <p type="main"> <s><emph type="italics"/>Tiri&longs;i la FG fin al centro del mondo, che &longs;ia in S, & dal punto S tiri&longs;i anco vna linea, <lb/>che tocchi il cerchio AFBG. non potrà già questa linea tirata dal punto S toc­<lb/>care il cerchio nel punto A; imperoche tirata la linea AS, il triangolo ACS ver<emph.end type="italics"/> <pb pagenum="10"/><emph type="italics"/>rebbe ad hauere due angoli retti, cioè SAC, & ACS, che è impoßibile: ne me <lb/>no toccherà &longs;opra il punto A nella circonferenza AF; peroche &longs;egherebbe il cer-<emph.end type="italics"/> 2 <arrow.to.target n="note30"></arrow.to.target><lb/><emph type="italics"/>chio. </s> <s>Toccherà dunque &longs;otto, & &longs;ia SO: &longs;iano dapoi congiunte le lince SD SL, <lb/>lequali &longs;eghino la circonferenza AOG ne' punti<emph.end type="italics"/> K<emph type="italics"/>H, & &longs;iano ancho congiunte le <lb/>linee C<emph.end type="italics"/>K <emph type="italics"/>CH. </s> <s>Et percioche il pe&longs;o, quanto egli è piu da pre&longs;&longs;o di F, tanto piu an­<lb/>co &longs;tà &longs;opra il centro; come il pe&longs;o in D preme, & &longs;tà piu &longs;opra il punto del volgi­<lb/>mento C, come à centro, cioè in D piu graua &longs;opra la linea CD, che &longs;e egli fo&longs;&longs;e in A <lb/>&longs;opra la linea CA: & dauantaggio piu in L &longs;opra la linea CL. </s> <s>imperoche e&longs;&longs;endo <lb/>li tre angoli di cia&longs;cun triangolo eguali à due angoli retti, & l'angolo DC<emph.end type="italics"/>K <emph type="italics"/>del <lb/>triangolo DC<emph.end type="italics"/>K, <emph type="italics"/>che è di due lati eguali &longs;ia <lb/>minore dell'angolo LCH del <expan abbr="triãgolo">triangolo</expan> LCH, <lb/>che è pur di due lati eguali: &longs;aranno gli altri <lb/>alla ba&longs;e, cioè CD<emph.end type="italics"/>K <emph type="italics"/>C<emph.end type="italics"/>K<emph type="italics"/>D in&longs;ieme pre&longs;i <lb/>maggiori de gli altri CLH CHL; & le <lb/>metà di que&longs;ti, cioè l'angolo CDS &longs;arà mag <lb/>giore dell'angolo CLS. </s> <s>E&longs;&longs;endo adunque <lb/>CLS minore, la linea CL piu &longs;i acco&longs;terà <lb/>al mouimento naturale del pe&longs;o me&longs;&longs;o in L <lb/>del tutto &longs;ciolto; cioè à dire alla linea LS, <lb/>che CD al mouimento DS: percioche il pe <lb/>&longs;o po&longs;to in L libero, & &longs;ciolto &longs;i mouerebbe <lb/>ver&longs;o il centro del mondo per LS, & il pe­<lb/>&longs;o po&longs;to in D per DS. </s> <s>Ma perche il pe&longs;o <lb/>me&longs;&longs;o in L graua tutto &longs;opra LS, & quello <lb/>che è in D &longs;opra DS, il pe&longs;o in L grauerà <lb/>pu &longs;opra la linea CL, che quello, che &longs;tà in <lb/>D &longs;opra la linea DC. </s> <s>Adunque la linea <lb/>CL &longs;o&longs;terrà piu il pe&longs;o, che lalinea CD, & <lb/>nel modo iste&longs;&longs;o quanto piu il pe&longs;o &longs;arà da <lb/>pre&longs;&longs;o ad F, &longs;i dimo strerà piu e&longs;&longs;er &longs;o&longs;tenuto <lb/>dalla linea CL per cotesta cagione, peroche <lb/>&longs;empre l'angolo CLS &longs;arebbe minore, la­<lb/>qual co&longs;a etiandio èmanife&longs;ta; perche &longs;e le li <lb/>nee CL, & LS s'incontra&longs;&longs;ero in vna li <lb/>nea, ilche auiene in FCS, all'hora la linea <lb/>CF &longs;o&longs;terrebbe tutto il pe&longs;o, che è in F, & <lb/>lo renderebbe immobile, nè haurebbe niuna <lb/>grauezza in tutto nella circonferenza del cer <lb/>chio. </s> <s>Li&longs;te&longs;&longs;o pe&longs;o dunque per la diuer&longs;ità<emph.end type="italics"/><lb/> <arrow.to.target n="fig12"></arrow.to.target><lb/><emph type="italics"/>de' &longs;iti &longs;arà piu graue, & piu lieue. </s> <s>& que&longs;to non già percio che per ragione del &longs;ito <lb/>alcuna volta egli acqui&longs;ti veramente grauezza maggiore, & alcuna volta la perda, <lb/>e&longs;&longs;endo &longs;empre della i&longs;te&longs;&longs;a grauezza, troui&longs;i douunque &longs;i voglia: ma percioche egli<emph.end type="italics"/> <pb/><emph type="italics"/>graua piu, & meno nella circonferenza, come in D piu graua &longs;opra la circonferenza <lb/>DA, che in L &longs;opra la circonferenza LD: cioè &longs;e il pe&longs;o &longs;arà &longs;o&longs;tenuto dalle circon <lb/>ferenze, & dalle linee diritte; la circonferenza AD &longs;o&longs;terrà piu il pe&longs;o po&longs;to in D, <lb/>che la circonferenza DL, &longs;tando il pe&longs;o in L; peroche meno aiuta CD, che CL. </s> <s><lb/>Oltre à ciò quando il pe&longs;o è in L, &longs;e egli fo&longs;&longs;e del tutto libero & &longs;ciolto, &longs;i mouerebbe <lb/>in giu per LS, &longs;e non gliene fu&longs;&longs;e vietato dalla linea CL, laquale sforza il pe&longs;o po&longs;to <lb/>in L à mouer&longs;i oltre la linea LS per la circonferenza LD, & lo caccia in certo mo <lb/>do, & in cacciandolo viene in parte à &longs;o&longs;tenerlo; percioche &longs;e non lo &longs;o&longs;tene&longs;&longs;e, & <lb/>gli face&longs;&longs;e re&longs;i&longs;tenza, &longs;i mouerebbe in giu per la linea LS, ma non già per la cir­<lb/>conferenza LD. </s> <s>Similmente la CD fa re&longs;i&longs;tenza al pe&longs;o po&longs;to in D, sforzan­<lb/>dolo à mouer&longs;i per la circonferenza DA. </s> <s>Nell'iste&longs;&longs;o modo &longs;tando il pe&longs;o in A, <lb/>la linea CA con&longs;tringerà il pe&longs;o à mouer&longs;i <lb/>oltre la linea AS per la circoferenza AO; <lb/>peroche l'angolo CAS è acuto, e&longs;&longs;endo lo <lb/>angolo ACS retto. </s> <s>Adunque le linee <lb/>CA CD in qualche parte, ma non già e­<lb/>gualmente fanno re&longs;istenza al pe&longs;o. </s> <s>& qua <lb/>lunque volta l'angolo, che è nella circonfe­<lb/>renza del cerchio fatto dalle linee che e&longs;cono <lb/>dal centro del monde S, & dal centro C &longs;a­<lb/>rà acuto, dimo&longs;treremo auenire l'i&longs;te&longs;&longs;o. </s> <s>Hor <lb/>percioche l'angolo mi&longs;to CLD è eguale à <lb/>l'angolo CDA, per e&longs;&longs;ere conteuuto da <lb/>mezi diametri, & dall'i&longs;te&longs;&longs;a circonferenza; <lb/>& l'angolo CLS è minore dell'angolo <lb/>CDS; &longs;arà il reflante SLD maggiore <lb/>del re&longs;tante SDA. </s> <s>Per laqual co&longs;a la cir <lb/>conferenza DA, cioè la di&longs;ce&longs;a del pe&longs;o <lb/>in D &longs;ara piu da pre&longs;&longs;o al mouimento natu­<lb/>rale del pe&longs;o &longs;ciolto me&longs;&longs;o in D, cioè della li­<lb/>nea DS, che la circonferenza LD della <lb/>linea LS. </s> <s>Meno dunque farà re&longs;i&longs;tenza la <lb/>linea CD al pe&longs;o po&longs;to in D, che la linea <lb/>CL al pe&longs;o po&longs;to in L. </s> <s>Però la linea CD <lb/>&longs;o&longs;terrà meno, che CL, & il pe&longs;o &longs;arà <lb/>piu libero in D, che in L: mouendo&longs;i piu <lb/>naturalmente il pe&longs;o per DA, che per LD. </s> <s><lb/>Per laqual co&longs;a piu graue &longs;arà in D, che in <lb/>L. </s> <s>Similmente dimo&longs;treremo, che CA man <lb/>co&longs;o&longs;tiene, che CD & che il pe&longs;o piu in A, <lb/>che in D è libero, & piu graue. </s> <s>Dopo dalla<emph.end type="italics"/><lb/> <arrow.to.target n="fig13"></arrow.to.target><lb/><emph type="italics"/>parte di &longs;otto per l'i&longs;te&longs;&longs;e cagioni, quanto il pe&longs;o &longs;arà piu da pre&longs;&longs;o al G, &longs;arà piu ri-<emph.end type="italics"/> <pb pagenum="11"/><emph type="italics"/>tenuto, come in H dalla linea CH, che in<emph.end type="italics"/> K <emph type="italics"/>dalla linea C<emph.end type="italics"/>K: <emph type="italics"/>percioche e&longs;&longs;en<emph.end type="italics"/> <arrow.to.target n="note31"></arrow.to.target><lb/><emph type="italics"/>do l'angolo CHS maggiore dell'angolo C<emph.end type="italics"/>K<emph type="italics"/>S, le linee CH HS, &longs;i acco&longs;te­<lb/>ranno piu alla direttione, che C<emph.end type="italics"/>K K<emph type="italics"/>S. </s> <s>& per que&longs;to &longs;arà piu ritenuto il pe&longs;o da <lb/>CH, che da C<emph.end type="italics"/>K; <emph type="italics"/>percioche &longs;e CH HS &longs;i incontra&longs;&longs;ero in vna linea, come auie­<lb/>ne &longs;tando il pe&longs;o in G, allbora la linea CG &longs;o&longs;terrebbe tutto il pe&longs;o in G, per <lb/>modo che &longs;tarebbe immobile. </s> <s>Quanto minore dunque &longs;arà l'angolo contenuto dal <lb/>la linea CH, & dalla di&longs;ce&longs;a del pe&longs;o &longs;ciolto, cioè dalla linea HS, tanto meno <lb/>anco quella linea ritenirà il pe&longs;o, & doue &longs;arà manco ritenuto, iui &longs;arà piu libero, & <lb/>piu graue. </s> <s>Oltre à ciò &longs;e il pe&longs;o fo&longs;&longs;e libero in K, & &longs;ciolto, &longs;i mouerebbe per la li­<lb/>nea KS, ma egli è impedito dalla linea CK, laquale sforza il pe&longs;o a mouer&longs;i di <lb/>qua dalla linea KS per la circonferenza KH; percio che lo ritira in certo modo, <lb/>& in ritirandolo viene a &longs;o&longs;tenerlo, peroche &longs;e non lo &longs;o&longs;tene&longs;&longs;e, &longs;i mouerebbe il pe­<lb/>&longs;o in giu per la linea diritta KS, ma non per la circonferenza KH. </s> <s>Similmente <lb/>la CH ritiene il pe&longs;o, sforzandolo a mouer&longs;i per la circonferenza HG. </s> <s>Et percio­<lb/>che l'angolo CHS è maggiore dell'angolo CKS, leuati via gli angoli eguali <lb/>CHG, C<emph.end type="italics"/>K<emph type="italics"/>H, &longs;arà il re&longs;tante SHG maggiore del re&longs;tante S<emph.end type="italics"/>K<emph type="italics"/>H. </s> <s>Adunque <lb/>la circonferenza KH, cioè la di&longs;ce&longs;a del pe&longs;o po&longs;to in K &longs;arà piu da pre&longs;&longs;o al mo­<lb/>uimento naturale del pe&longs;o po&longs;to in K &longs;ciolto, cioè alla linea KS, che la circonfe­<lb/>renza HG alla linea HS. </s> <s>Per laqual co&longs;a meno ritiene la linea CK, che CH, <lb/>mouend o&longs;i il pe&longs;o piu naturalmente per KH, che per HG, Con ragione &longs;imile <lb/>anco &longs;i mo&longs;trerà, che quanto minore &longs;arà l'angolo SKH, la linea CK &longs;o&longs;terrà <lb/>meno. </s> <s>Stando dunque il pe&longs;o in O, percioche l'angolo SOC non &longs;olamente è <lb/>minore dell'angolo CKS, ma anco il minimo di tutti gli angoli, che e&longs;con da i pun <lb/>ti CS, & hanno la cima nella circonferenza OKG; &longs;arà l'angolo SOK il mi <lb/>nimo &longs;i dell'angolo SKH, come de tutti gli altri co&longs;i fatti. </s> <s>Adunque la di&longs;ce&longs;a <lb/>del pe&longs;o po&longs;to in O &longs;arà piu da pre&longs;&longs;o al mouimento naturale di e&longs;&longs;o pe&longs;o &longs;ciolto in <lb/>O, che in altro &longs;ito della circonferenza OKG: & la linea CO meno &longs;o&longs;tenirà <lb/>il pe&longs;o, che &longs;e egli fo&longs;&longs;e in qual &longs;i voglia altro &longs;ito della i&longs;te&longs;&longs;a circonferenza OG. </s> <s><lb/>Similmente perche l'angolo del toccamento SOK è minore &longs;i dell'angolo SDA, <lb/>&longs;i dello SAO, & &longs;i di qual &longs;i voglia altro &longs;imile; &longs;arà la &longs;ce&longs;a del pe&longs;o me&longs;&longs;o in O <lb/>piu da pre&longs;&longs;o al mouimento naturale di e&longs;&longs;o pe&longs;o &longs;ciolto in O, che in altro &longs;ito del­<lb/>la <expan abbr="circõfer&etilde;za">circonferenza</expan> ODF. </s> <s>Oltre a ciò perche la linea CO no puote &longs;pingere il pe&longs;o po&longs;to <lb/>in O mentre egli &longs;i moue in giu, per modo che egli &longs;i moua oltre la linea OS, per <lb/>cioche la linea OS non taglia il cerchio, ma lo tocca; & l'angolo SOC è retto <lb/>& non acuto, il pe&longs;o po&longs;to in O non grauerà niente &longs;opra la linea CO, ne &longs;tarà <lb/>&longs;opra il centro, come accaderebbe in qual &longs;i voglia altro punto &longs;opra l'O. </s> <s>Sarà dun­<lb/>que il pe&longs;o po&longs;to in O per que&longs;te cagioni libero, & &longs;ciolto piu in que&longs;to &longs;ito, che in <lb/>qual &longs;i voglia altro della circonferenza FOG; & perciò in que&longs;to &longs;arà piu graue, <lb/>cioè a dire piu grauerà, che in altro &longs;ito. </s> <s>Et quanto &longs;arà piu da pre&longs;&longs;o ad O, &longs;arà <lb/>piu graue di quello, che &longs;e fo&longs;&longs;e piu da lunge: & la linea CO &longs;arà egualmente di­<lb/>&longs;tante dall'orizonte: non pero all'orizonte del punto C (come &longs;timano e&longs;&longs;i) ma <lb/>del pe&longs;o po&longs;to in O, douendo&longs;i prendere l'orizonte dal centro della grauezza del pe <lb/>&longs;o. </s> <s>Lequali co&longs;e tutte bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <pb/> <p type="margin"> <s><margin.target id="note30"></margin.target><emph type="italics"/>Per la deci ma ottau<gap/> del terzo<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note31"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 21. <emph type="italics"/>del prim.<emph.end type="italics"/></s> </p> <figure id="fig12"></figure> <figure id="fig13"></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e il braccio della bilancia fo&longs;&longs;e maggiore <lb/>di CO, come per la quantità di CD; <lb/>&longs;arà parimente il pe&longs;o me&longs;&longs;o in O piu gra­<lb/>ue. </s> <s>De&longs;criua&longs;i il cerchio OH, il cui <lb/>centro &longs;ia D, & il mezo diametro D</s> <s>/>il cerchio OH toccherà il cerchio FOG <lb/>nel punto O, & toccherà anche la linea<emph.end type="italics"/><lb/> <arrow.to.target n="note32"></arrow.to.target> <emph type="italics"/>OS nel punto mede&longs;imo, laquale è la &longs;ce­<lb/>&longs;a naturale, & diritta del pe&longs;o po&longs;to in O.<emph.end type="italics"/><lb/> <arrow.to.target n="note33"></arrow.to.target> <emph type="italics"/>Et percioche l'angolo SOH è minore del <lb/>l'angolo SOG, &longs;arà la &longs;ce&longs;a del pe&longs;o po&longs;to <lb/>in O per la circonferenza OH piu dapre&longs; <lb/>&longs;o al mouimento naturale OS, che per la <lb/>circonferenza OG. </s> <s>Piu libero dunque <lb/>& &longs;ciolto, & per con&longs;equente piu graue &longs;a­<lb/>ràin O, &longs;tante il centro della bilancia in <lb/>D, che in C. </s> <s>Similmente &longs;i mo&longs;trerà, <lb/>che quanto piu grande &longs;arà il braccio DO, <lb/>il pe&longs;o po&longs;to in O &longs;arà d'auantaggio piu <lb/>graue.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note32"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del terzo<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note33"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del terzo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e l'iste&longs;&longs;o cerchio AFBG co'l &longs;uo centro R &longs;arà piu da pre&longs;&longs;o ad S centro <lb/>del mondo, & dal punto S &longs;ia tirata vna linea, che tocchi il cerchio ST, il pun­<lb/>to T, (doue il pe&longs;o è piu graue) &longs;arà piu lontano dal punto A, che il punto O: <lb/>percioche &longs;iano tirate da i punti OT le linee OMTN à piombo di CS, & <lb/>congiungan &longs;i RT, & &longs;ia il centro R nella linea CS, & la linea ARB &longs;ia <lb/>egualmente di&longs;tante ad ACB. </s> <s>Percioche dunque i triangoli COS RTS &longs;ono<emph.end type="italics"/><lb/> <arrow.to.target n="note34"></arrow.to.target> <emph type="italics"/>di angoli retti, &longs;arà SC à CO, come CO à CM. </s> <s>Similmente SR ad<emph.end type="italics"/><lb/> <arrow.to.target n="note35"></arrow.to.target> <emph type="italics"/>RT, come RT ad RN. </s> <s>E&longs;&longs;endo dunque RT eguale à CO, & SC mag <lb/>giore di RS: haurà proportione maggiore SC à CO, che SR ad RT. </s> <s>on<emph.end type="italics"/><lb/> <arrow.to.target n="note36"></arrow.to.target> <emph type="italics"/>de baurà parimente proportione maggiore CO à CM, che RT ad RN. </s> <s>&longs;a <lb/>rà dunque minore CM, che RN. </s> <s>Tagli&longs;i dunque RN in P &longs;i fattamen-<emph.end type="italics"/> <pb pagenum="12"/><emph type="italics"/>te, che RP &longs;ia eguale à CM; & dal <lb/><expan abbr="pũto">punto</expan> P &longs;ia tirata la linea PQ egual <lb/>mente di&longs;tante dalle linee MONT, <lb/>laquale tagli la <expan abbr="circõfer&etilde;za">circonferenza</expan> AT in Q, <lb/>& in fine <expan abbr="cõgiõgan&longs;i">congiongan&longs;i</expan> la <expan abbr="Rq.">Rque</expan> Hor per <lb/>cioche le due CO CM &longs;ono eguali à <lb/>le due RQ RP, & l'angolo CMO<emph.end type="italics"/> <arrow.to.target n="note37"></arrow.to.target><lb/><emph type="italics"/>è eguale all'angolo <expan abbr="RPq;">RPque</expan> &longs;arà an­<lb/>che l'angolo MCO eguale all'angolo <lb/><expan abbr="PRq.">PRque</expan> Ma l'angolo MCA retto <lb/>è eguale all'angolo PRA retto; a­<lb/>dunque il re&longs;tante OCA al restante<emph.end type="italics"/> <arrow.to.target n="note38"></arrow.to.target><lb/><emph type="italics"/>QRA &longs;arà eguale, & la circonferen­<lb/>za OA parimente eguale alla circon <lb/>ferenza QA. </s> <s>Però il punto T per <lb/>e&longs;&longs;ere piu di&longs;tante dal punto A, che <lb/>Q, &longs;arà anco piu di&longs;tante dal punto <lb/>A, che il punto O. </s> <s>Dimo&longs;trera&longs;&longs;i pa <lb/>rimente, che quanto piu il cerchio &longs;arà <lb/>vicino al centro del mondo, che egli &longs;a <lb/>rà anco piu lontano. </s> <s>Et co&longs;i come pri­<lb/>ma dimo&longs;trera&longs;&longs;i il pe&longs;o nella cir confe­<lb/>renza TAF &longs;tar &longs;opra il centro R, <lb/>ma nella circonferenza TG e&longs;&longs;ere ri­<lb/>tenuto dalla linea, & ritrouar&longs;i piu gra <lb/>ue nel punto T.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note34"></margin.target><emph type="italics"/>Per la <gap/>ttaua del &longs;esto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note35"></margin.target><emph type="italics"/>Per la ottaua del quito<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note36"></margin.target><emph type="italics"/>Per la decima del q<gap/> to.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note37"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del &longs;esto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note38"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 26. <emph type="italics"/>del terzo.<emph.end type="italics"/></s> </p> <figure></figure> <pb/> <p type="main"> <s><emph type="italics"/>Che &longs;e il punto G fo&longs;&longs;e nel centro del mondo; allhora quanto piu il pe&longs;o &longs;arà da pre&longs;&longs;o al <lb/>G, &longs;arà piu graue: & douunque &longs;ia po&longs;to il pe&longs;o, fuor che nel G &longs;empre &longs;tarà &longs;opra <lb/>il centro C, come in<emph.end type="italics"/> K<emph type="italics"/>: Imperoche tirata la linea G<emph.end type="italics"/>K; <emph type="italics"/>que&longs;ta (&longs;e condo laqua <lb/>le &longs;i fa il mouimento naturale del pe&longs;o) in&longs;ieme co'l braccio della bilancia<emph.end type="italics"/> K<emph type="italics"/>C <lb/>farà vn'angolo acuto, peroche <lb/>gli angoli posti alla ba&longs;e in<emph.end type="italics"/> K <lb/><emph type="italics"/>& G del triangolo di due la <lb/>ti eguali C<emph.end type="italics"/>K<emph type="italics"/>G &longs;ono &longs;empre <lb/>acuti. </s> <s>Hor &longs;iano paragonate <lb/>in&longs;ieme que&longs;te due co&longs;e, cioè il <lb/>pe&longs;o posto in<emph.end type="italics"/> K, <emph type="italics"/>& quello, <lb/>che è po&longs;to in D, &longs;arà il pe&longs;o <lb/>in K piu graue, che quello <lb/>in D; imperoche tirata la li­<lb/>nea DG, e&longs;&longs;endo che li tre an <lb/>goli di cia&longs;cuno triangolo &longs;iano <lb/>eguali à due angoli retti, & <lb/>l'angolo DCG del triangolo <lb/>CDG di due lati eguali &longs;ia <lb/>maggiore dell'angolo KCG <lb/>del triangolo CKG di due <lb/>lati eguali; &longs;aranno gli altri an <lb/>goli alla ba&longs;e DGC GDC <lb/>pre&longs;i in&longs;ieme minori de gli al­<lb/>tri KGC GKC pre&longs;i in&longs;ie<emph.end type="italics"/><lb/> <arrow.to.target n="fig14"></arrow.to.target><lb/><emph type="italics"/>me; & la metà di questi, cioè l'angolo CDG &longs;arà minore dell'angolo CKG: <lb/>Per laqual co&longs;a mouendo&longs;i il pe&longs;o po&longs;to in K &longs;ciolto naturalmente per KG, & <lb/>il pe&longs;o po&longs;to in D per DG come per &longs;patij, per i quali &longs;ono portati nel centro del <lb/>mondo; la linea CD, cioè il braccio della bilancia &longs;i acco&longs;terà piu al mouimento <lb/>naturale del pe&longs;o po&longs;to in D <expan abbr="totalm&etilde;te">totalmente</expan> &longs;ciolto, alla linea cioè DG, che CK al <lb/>mouimento &longs;atto &longs;econdo KG. </s> <s>So&longs;tenterà dunque piu la linea CD, che C</s> <s>/>& perciò il pe&longs;o po&longs;to in K per le co&longs;e di &longs;opra dette &longs;arà piu graue, che in D. </s> <s>Ol­<lb/>tre à ciò, perche &longs;e il pe&longs;o po&longs;to in K fo&longs;&longs;e del tutto libero, & &longs;ciolto, &longs;i mouerebbe <lb/>in giu per KG, &longs;e egli non fo&longs;&longs;e impedito dalla linea CK, laquale sforza il pe&longs;o <lb/>à mouer&longs;i oltra la linea KG per la circonferenza KH; la linea KG &longs;o&longs;tente­<lb/>rà il pe&longs;o in parte, & gli farà re&longs;istenza, sforzandolo à mouer&longs;i per la circonferenza <lb/>KH. </s> <s>Et percioche l'angolo CDG è minore dell'angolo CKG, & l'angolo <lb/>CDK è eguale all'angolo CKH, &longs;arà l'angolo re&longs;tante GDK maggiore del re <lb/>&longs;tante GKH. </s> <s>Dunque la circonferenza KH &longs;arà piu da pre&longs;&longs;o al mouimento <lb/>naturale del pe&longs;o &longs;ciolto po&longs;to in K, cioè alla linea KG, che la circonferenza <lb/>DK alla linea DG. </s> <s>Per laqual co&longs;a la linea CD &longs;a piu re&longs;i&longs;tenza al pe&longs;o po&longs;to <lb/>in D, che la linea CK al pe&longs;o posto in K. </s> <s>Adunque il pe&longs;o po&longs;to in K &longs;arà<emph.end type="italics"/> <pb pagenum="13"/><emph type="italics"/>piu graue, che in D. </s> <s>Similmente mostrera&longs;&longs;i, che quanto il pe&longs;o &longs;arà piu da pre&longs;&longs;o <lb/>ad F, come in L manco grauerà; ma quanto piu da pre&longs;&longs;o &longs;i trouerà al G, co­<lb/>me in H, e&longs;&longs;ere piu graue.<emph.end type="italics"/></s> </p> <figure id="fig14"></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e il centro del mondo fo&longs;&longs;e in S fra i punti CG; Primieramente &longs;i mo&longs;trerà nel <lb/>modo i&longs;te&longs;&longs;o, che il pe&longs;o in qualunque luogo po&longs;to starà &longs;opra il centro C, come in <lb/>H: peroche tirate le li­<lb/>nee HG HS, l'angolo <lb/>che è alla ba&longs;e GHC del <lb/><expan abbr="triãgolo">triangolo</expan> di due lati eguali <lb/>CHG è &longs;empre acuto: <lb/>Perlaqual co&longs;a anco SHC <lb/>minor di lui &longs;arà parimen <lb/>te &longs;empre acuto. </s> <s>ma &longs;ia ti <lb/>rata dal punto S la linea <lb/>SK à piombo di CS. </s> <s><lb/>Dico che il pe&longs;o è piu gra­<lb/>ue in<emph.end type="italics"/> K, <emph type="italics"/>che in alcun'al <lb/>tro &longs;ito della circonferen <lb/>za FKG; & quanto <lb/>piu da pre&longs;&longs;o &longs;arà allo F, <lb/>ouero al G meno graue­<lb/>rà. </s> <s>Prendan&longs;i ver&longs;o lo <lb/>F i punti DL, & con <lb/><expan abbr="giungã&longs;i">giungan&longs;i</expan> le linee LC LS <lb/>DC DS, & &longs;iano al-<emph.end type="italics"/><lb/> <arrow.to.target n="fig15"></arrow.to.target><lb/><emph type="italics"/>lungate le linee LS DS KS HS fin'alla <expan abbr="circõferenza">circonferenza</expan> del cerchio in EM NO; <lb/>& &longs;iano <expan abbr="cõgiunte">congiunte</expan> CE, CM, CN, CO. </s> <s>Hor percioche LE DM &longs;i taglia­<lb/><gap/>o in&longs;ieme in S, &longs;arà il rettangolo LSE eguale al rettangolo DSM. </s> <s>Onde &longs;i co<emph.end type="italics"/> <arrow.to.target n="note39"></arrow.to.target><lb/><emph type="italics"/>me è la LS ver&longs;o la DS, co&longs;i &longs;arà la SM ver&longs;ola SE; ma è maggior la LS <lb/>della DS; & la SM di e&longs;&longs;a SE. </s> <s>Dunque LS SE pre&longs;e in&longs;ieme &longs;aranno mag-<emph.end type="italics"/> <arrow.to.target n="note40"></arrow.to.target><lb/><emph type="italics"/>giori delle DS SM. </s> <s>& per la ragion i&longs;te&longs;&longs;a &longs;i mo&longs;trerà la KN e&longs;&longs;er minore di DM. </s> <s><lb/>Di piu percioche il rettangolo OSH è eguale alrett'angolo KSN; per la mede&longs;i-<emph.end type="italics"/> <arrow.to.target n="note41"></arrow.to.target><lb/><emph type="italics"/>ma ragione la HO &longs;arà maggiore della KN. & nell'i&longs;te&longs;&longs;o modo in tutto la<emph.end type="italics"/> <arrow.to.target n="note42"></arrow.to.target><lb/><emph type="italics"/>KN &longs;i dimostrerà minore di tutte le altre linee, che pa&longs;&longs;ino per lo punto S. </s> <s>Et <lb/>percioche de i triangoli di due lati eguali CLE DCM i lati LC CE &longs;ono e­<lb/>guali a i lati DC CM; & la ba&longs;e LE è maggiore di DM: &longs;arà l'angolo <lb/>LCE maggiore dell'angolo DCM. </s> <s>Per laqual co&longs;a gli angoli CLE CEL po<emph.end type="italics"/> <arrow.to.target n="note43"></arrow.to.target><lb/><emph type="italics"/>sti alla ba&longs;e tolti in&longs;ieme &longs;aranno minori de gli angoli CDM CMD; & le me­<lb/>tà di que&longs;ti, cioè l'angolo CLS &longs;arà minore dell'angolo CDS. </s> <s>Dunque il pe&longs;o po <lb/>&longs;to in L &longs;opra la linea LC grauerà piu, che po&longs;to in D &longs;opra la DC; & piu <lb/>&longs;tarà &longs;opra il centro in L, che in D. </s> <s>Similmente &longs;i mo&longs;trerà, che il pe&longs;o in D<emph.end type="italics"/> <pb/><emph type="italics"/>&longs;tarà piu &longs;opra il centro C, che in K. </s> <s>Adunque il pe&longs;o po&longs;to in K &longs;arà piu <lb/>graue, che in D, & in D, che in L. & con la mede&longs;ima ragione in tutto, pero­<lb/>che KN è minore di HO, &longs;arà l'angolo CKS maggiore dell'angolo CHS. </s> <s><lb/>Per laqual cofa il pe&longs;o posto in H &longs;tarà piu &longs;opra il centro C, che in<emph.end type="italics"/> K; <emph type="italics"/>& in que­<lb/>&longs;ta maniera &longs;i mostrerà, che douunque &longs;ia il pe&longs;o nella circonferenza FDG, manco <lb/>starà &longs;opra il centro quando &longs;arà po&longs;to in K, che in altro &longs;ito: & quanto piu da <lb/>pre&longs;&longs;o egli &longs;arà ad F, ouero à G piu &longs;tarà &longs;opra. </s> <s>Dopo percioche l'angolo CKS <lb/>è maggiore del CDS, & CDK è eguale à CKH: &longs;arà il re&longs;tante SKH mi­<lb/>nore del re&longs;tante SDK. </s> <s>Per laqual co&longs;a la circonferenza KH &longs;arà piu da pre&longs;&longs;o <lb/>al mouimento naturale <lb/>diritto del pe&longs;o po&longs;to in <lb/>K &longs;ciolto, cioè alla li­<lb/>nea KS, che la circon <lb/>ferenza DK al moui­<lb/>mento DS. </s> <s>& perciò <lb/>la linea CD &longs;a piu re&longs;i <lb/>&longs;tenza al pe&longs;o po&longs;to in D <lb/>che la CK al pe&longs;o me&longs;­<lb/>&longs;o in<emph.end type="italics"/> K. <emph type="italics"/>& per que&longs;ta <lb/>ragione &longs;i mo&longs;trera l'an­<lb/>golo SHG e&longs;&longs;er mag­<lb/>giore dello SKH; & <lb/>per con&longs;equente la linea <lb/>CH &longs;are piu re&longs;i&longs;tenza <lb/>al pe&longs;o po&longs;to in H, che <lb/>CK al pe&longs;o me&longs;&longs;o in K. </s> <s><lb/>Similmente dimo&longs;trera&longs;&longs;i <lb/>che la linea CL piu &longs;o­<lb/>&longs;tenterà il pe&longs;o, che CD:<emph.end type="italics"/><lb/> <arrow.to.target n="fig16"></arrow.to.target><lb/><emph type="italics"/>& per le, cagioni i&longs;te&longs;&longs;e &longs;i prouerà, che il pe&longs;o me&longs;&longs;o in K grauerà meno &longs;opra la li­<lb/>nea CK, che in qual &longs;i voglia altro &longs;ito della circon&longs;erenza FDG: & quanto <lb/>piu da pre&longs;&longs;o &longs;arà ad F, ouero à G, manco grauerà. </s> <s>dunque piu graue &longs;ara in K, <lb/>che in altro &longs;ito: & &longs;arà meno graue quanto piu da pre&longs;&longs;o &longs;tara ad F, ouero a G.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note39"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 35. <emph type="italics"/>del terzo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note40"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del &longs;esto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note41"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del terzo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note42"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 25. <emph type="italics"/>del quinte.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note43"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 25. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure id="fig15"></figure> <figure id="fig16"></figure> <p type="main"> <s><emph type="italics"/>Se in fine il centro C fo&longs;&longs;e nel centro del mondo, egli è manife&longs;to, che il pe&longs;o po&longs;to doue<emph.end type="italics"/><lb/> <arrow.to.target n="note44"></arrow.to.target> <emph type="italics"/>&longs;i voglia &longs;tarà fermo. </s> <s>Come posto il pe&longs;o in D la linea CD &longs;o&longs;terrà tutto il pe&longs;o, <lb/>per e&longs;&longs;er a piombo dell'orizonte di e&longs;&longs;o pe&longs;o po&longs;to in D. </s> <s>Dunque &longs;tarà fermo <lb/>il pe&longs;o.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note44"></margin.target><emph type="italics"/>Per la pri<gap/>ua di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Hor percioche nelle co&longs;e, che fin qui &longs;ono &longs;tate dimostrate non habbiamo fatto mentio­<lb/>ne alcuna della grauezza del braccio della bilancia, però &longs;e vorremo anco con&longs;idera­<lb/>re la grauezza del detto braccio, &longs;i potrà ritrouare il centro della grauezza della ma<emph.end type="italics"/> <pb pagenum="14"/><emph type="italics"/>gnitudine fatta dal pe&longs;o, & dal braccio, & &longs;i <expan abbr="de&longs;criuerãno">de&longs;criueranno</expan> le circonferenze de' cerch; <lb/>&longs;econdo la di&longs;tanza dal centro della bilancia ad e&longs;&longs;o centro della grauezza, come &longs;e <lb/>in e&longs;&longs;o (come è veramente) fo&longs;&longs;e posto il pe&longs;o, Et le co&longs;e che &longs;enza la con&longs;ideratio <lb/>ne della grauezza del braccio della bilancia habbiamo trouato, tutte nell'i&longs;te&longs;&longs;o mo­<lb/>do con&longs;iderando ancora tal grauit à le ritrouaremo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Dalle co&longs;e dette dunque, <expan abbr="con&longs;iderãdo">con&longs;iderando</expan> la bilancia, <lb/>come ella è lontana dal centro del mondo <lb/>nel modo che e&longs;&longs;i hanno fatto, come etiandio <lb/>è in atto, appare la fal&longs;it à di coloro, che dico­<lb/>no il pe&longs;o po&longs;to in A e&longs;&longs;ere piu graue, che <lb/>in altro &longs;ito; & in&longs;ieme e&longs;&longs;er fal&longs;o, che quan­<lb/>to piu il pe&longs;o è lontano dalla linea FG, tan­<lb/>to e&longs;&longs;ere piu graue: imperoche il punto O <lb/>è piu da pre&longs;&longs;o alla FG, che il punto A; <lb/>percioche la linea tirata a piombo dal pun-<emph.end type="italics"/> <arrow.to.target n="note45"></arrow.to.target><lb/><emph type="italics"/>to O ad FG è minore della CA. </s> <s>Da poi <lb/>egli è parimente fal&longs;o, che il pe&longs;o dal punto <lb/>A &longs;i moua piu velocemente, che da altro <lb/>&longs;ito. </s> <s>peroche dal punto O &longs;i mouerà piu ve <lb/>locemente, che dal punto A, concio&longs;ia che <lb/>in O &longs;ia piu libero e &longs;ciolto, che in altro &longs;ito; <lb/>& la &longs;ce&longs;a dal punto O &longs;ia piu da pre&longs;&longs;o al <lb/>mouimento naturale diritto, che qual &longs;i vo­<lb/>glia altra di&longs;ce&longs;a.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note45"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>del terzo.<emph.end type="italics"/></s> </p> <figure></figure> <pb/> <p type="main"> <s><emph type="italics"/>Oltre a ciò quando mo&longs;trano per via della piu diritta, & della piu torta di&longs;ce&longs;a, che il pe­<lb/>&longs;o è piu graue in A, che in D, & in D, che in L. </s> <s>Primier amente per certo e&longs;tima <lb/>no il fal&longs;o, che &longs;e alcun pe&longs;o &longs;arà collocato in qual &longs;i voglia &longs;ito della circonferenza, <lb/>come in D, la &longs;ua vera di&longs;ce&longs;a douer&longs;i fare per la linea diritta DR egualmente di­<lb/>&longs;tante da e&longs;&longs;a FG, come &longs;econdo il mouimento naturale, &longs;i come prima è &longs;tato det­<lb/>to. </s> <s>Percioche in qual &longs;i voglia &longs;ito &longs;i collochi alcun pe&longs;o, &longs;e riguardiamo il mouimen <lb/>to &longs;uo naturale al proprio luogo, alquale &longs;i moue dirittamente per &longs;ua natura, pre&longs;up <lb/>po&longs;ta tutta la figura dell'vniuer&longs;o mondo, &longs;arà tale, che &longs;empre lo &longs;patio, per lo qua­<lb/>le &longs;i moue naturalmente, parerà hauere ragione di linea tirata dalla circonferenza al <lb/>centro. </s> <s>Adunque le na <lb/>turali di&longs;ce&longs;e diritte di <lb/>qual &longs;i voglia pe&longs;o &longs;ciol <lb/>to non &longs;i po&longs;&longs;ono fare <lb/>per linee tra loro egual <lb/>mente di&longs;tanti, per an­<lb/>dar&longs;i à trouar tutte nel <lb/>centro del mondo. </s> <s>pre <lb/>&longs;uppongono da poi, che <lb/>il pe&longs;o mo&longs;&longs;o da D in <lb/>A per linea diritta ver <lb/>&longs;o il centro del mondo <lb/>&longs;ia della <expan abbr="quãtità">quantità</expan> i&longs;te&longs;&longs;a, <lb/>come &longs;e egli fo&longs;&longs;e da O <lb/>in C &longs;i fattamente, <lb/>che il <expan abbr="pũto">punto</expan> A &longs;ia egual <lb/>mente di&longs;tante dal cen­<lb/>tro del mondo, come C; <lb/>ilche è parimente fal&longs;o:<emph.end type="italics"/><lb/> <arrow.to.target n="fig17"></arrow.to.target><lb/> <arrow.to.target n="note46"></arrow.to.target> <emph type="italics"/>Imperoche il punto A è piu da lontano dal centro del mondo, che C: percioche <lb/>maggior è la linea tirata dal centro del mondo fin ad A, che quella del centro del <lb/>mondo fin a C, concio&longs;ia che vna linea dal centro del mondo fin ad A &longs;i di&longs;tenda <lb/>&longs;otto vn'angolo retto contenuto dalle linee AC, & dal punto C al centro del <lb/>mondo. </s> <s>Dalle quali co&longs;e non &longs;olo rie&longs;ce vana quella pre&longs;uppo&longs;ta, laquale dimostra, <lb/>che la bilancia DE ritorna in AB, ma anco cadono tutte le loro dimo&longs;trationi; <lb/>&longs;e for&longs;e non dice&longs;&longs;ero, che que&longs;te co&longs;e tutte per la grandi&longs;&longs;ima di&longs;tanza, che è fra il cen <lb/>tro del mondo, & noi &longs;ono co&longs;i in&longs;en&longs;ibili, che per cagione di que&longs;ta in&longs;en&longs;ibilità, <lb/>&longs;i po&longs;&longs;ano pre&longs;upponere, come vere; concio&longs;ia, che tutti quelli, iquali hanno trattato <lb/>que&longs;te co&longs;e, le habbiano pre&longs;uppo&longs;te, come note; ma&longs;&longs;imamente, percioche quello <lb/>e&longs;&longs;ere in&longs;en&longs;ibile non fà, che la di&longs;ce&longs;a del pe&longs;o da L in D (per v&longs;are le loro paro­<lb/>le) non pigli meno del diretto, che la di&longs;ce&longs;a DA. </s> <s>Similmente l'arco DA piglie­<lb/>rà piu del diretto, che la circon&longs;erenza EV. </s> <s>onde &longs;arà vera la pre&longs;uppo&longs;ta, & le <lb/>altre dimo&longs;trationi rimarranno nella &longs;ua &longs;ua forza. </s> <s>Concediamo etiandio, che il pe<emph.end type="italics"/> <pb pagenum="15"/><emph type="italics"/>&longs;o po&longs;to in A &longs;ia piu graue, che in altro &longs;ito; & che la di&longs;ce&longs;a diritta del pe&longs;o &longs;i deb <lb/>ba fare per linea diritta egualmente di&longs;tante da FG, & quali&longs;i voglian punti pre&longs;i <lb/>nelle linee egualmente di&longs;tanti dall'orizonte e&longs;&longs;ere egualmente lontani dal centro <lb/>del mondo: non &longs;eguiter à gia per que&longs;to, che la loro dimostratione &longs;ia vera, con la­<lb/>quale vengono a dire, che il pe&longs;o posto in A è piu grane, che in altro &longs;ito, come in <lb/>L. </s> <s>Percioche &longs;e egli fo&longs;&longs;e vero, che quanto piu il pe&longs;o in que&longs;ta maniera di&longs;cende <lb/>piu al diritto, iui fo&longs;&longs;e piu graue; &longs;eguirebbe etiandio, che quanto l'iste&longs;&longs;o pe&longs;o de­<lb/>&longs;cende&longs;&longs;e egualmente in archi eguali al diritto, che ne i luoghi mede&longs;imi haue&longs;&longs;e gra­<lb/>uezza eguale, ilche in que&longs;to modo e&longs;&longs;er fal&longs;o &longs;i dimo&longs;tra.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note46"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure id="fig17"></figure> <p type="main"> <s><emph type="italics"/>Siano le circonferenze AL AM tra loro eguali, & congiunga&longs;i LM, laquale ta­<lb/>gli AB in X; &longs;arà LM egualmente di&longs;tante da FG, & à piombo di AB,<emph.end type="italics"/> <arrow.to.target n="note47"></arrow.to.target><lb/><emph type="italics"/>& XM &longs;arà eguale ad XL. </s> <s>Se dunque il pe&longs;o da L &longs;arà mo&longs;&longs;o in A per la cir­<lb/>conferenza LA, il mouimento &longs;uo diritto &longs;arà &longs;econdo la linea LX. </s> <s>Ma &longs;e egli &longs;i <lb/>mouerà da A in M per la circonferenza AM, il &longs;uo mouimento &longs;arà &longs;econdo <lb/>la linea diritta XM. </s> <s>Per laqual co&longs;a la &longs;ce&longs;a da L in A &longs;arà eguale alla &longs;ce&longs;a da <lb/>A in M, &longs;i per cau&longs;a delle circonferenze eguali, & &longs;i per le linee rette eguali, & à <lb/>piombo di e&longs;&longs;a AB. </s> <s>Adunque il pe&longs;o mede&longs;imo po&longs;to in L grauerà egualmente, <lb/>come in A, ilche è fal&longs;o, concio&longs;ia, che egli è di gran lunga piu graue in A, che in L.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note47"></margin.target><emph type="italics"/>Per la terza del terzo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et benche AMLA prendano, &longs;econdo e&longs;&longs;i, egualmente del diretto, diranno for&longs;e, <lb/>nondimeno perche il principio della &longs;ce&longs;a da L, cioè LD piglia meno del diretto, che <lb/>il principio della &longs;ce&longs;a da A, cioè AN, il pe&longs;o in A &longs;arà piu graue, che in L. </s> <s><lb/>Imperoche e&longs;&longs;endo (come è &longs;tato di &longs;opra po&longs;to) la circonferenza AN eguale ad <lb/>LD, laquale (&longs;econdo eßi) piglia di diretto CT; ma LD piglia di diretto PO, <lb/>però il pe&longs;o &longs;arà piu graue in A, che in L. ilche &longs;e fo&longs;&longs;e vero, &longs;eguirebbe, che l'i&longs;te&longs;­<lb/>&longs;o pe&longs;o nel mede&longs;imo &longs;ito, in diuer&longs;o modo &longs;olamente con&longs;iderato, ver&longs;o il mede&longs;imo <lb/>&longs;ito fo&longs;&longs;e & piu graue, & piu lieue; ilche è impo&longs;&longs;ibile. </s> <s>cioè &longs;e con&longs;ideriamo la &longs;ce&longs;a <lb/>del pe&longs;o po&longs;to in L in quanto egli de&longs;cende da L in A &longs;arà piu graue, che &longs;e con&longs;ide <lb/>reremo la &longs;ce&longs;a del pe&longs;o i&longs;te&longs;&longs;o da L in D &longs;olamente. </s> <s>ne po&longs;&longs;ono negare per i mede <lb/>&longs;imi detti &longs;uoi, che la di&longs;ce&longs;a del pe&longs;o da L in A non pigli del diretto LX, ouero PC. </s> <s><lb/>Et che &longs;imilmente la &longs;ce&longs;a AM non prenda di diretto XM: pigliando cßi ancora <lb/>à que&longs;to modo, & co&longs;i nece&longs;&longs;ario &longs;ia di pigliare. </s> <s>percioche &longs;e vogliono dimo&longs;trare, <lb/>che la bilancia DE ritorni in AB paragonando la &longs;ce&longs;a del pe&longs;o po&longs;to in D con <lb/>la &longs;ce&longs;a del pe&longs;o posto in E, egli è nece&longs;&longs;ario, che mo&longs;trino, che la diritta &longs;ce&longs;a OC <lb/>ri&longs;pondente alla circonferenza DA &longs;ia maggiore della &longs;ce&longs;a diritta TH ri&longs;ponden­<lb/>te alla circonferenza EV. peroche &longs;e piglia&longs;&longs;ero &longs;olamente vna parte di tutta la &longs;ce <lb/>&longs;a da D in A, come D<emph.end type="italics"/>K, <emph type="italics"/>& dimo&longs;tra&longs;&longs;ero, che piu di diretto piglia la &longs;ce&longs;a D<emph.end type="italics"/>K, <lb/><emph type="italics"/>che la eguale portione della &longs;ce&longs;a dal punto E, &longs;eguirebbe il pe&longs;o po&longs;to in D, &longs;econ­<lb/>do eßi, e&longs;&longs;ere piu graue del pe&longs;o po&longs;to in E, & mouer&longs;i in giu fin al K &longs;olamente. </s> <s><lb/>per modo che la bilancia &longs;ia mo&longs;&longs;a in KI. </s> <s>Similmente &longs;e vogliono mo&longs;trare, che la <lb/>bilancia KI ritorni in AB pigliando vna portione della &longs;ce&longs;a da K in A, cioè KS, <lb/>& mo&longs;tra&longs;&longs;ero, che KS pigli piu di diretto, che la &longs;ce&longs;a eguale, che è dirimpetto dal <lb/>punto I: &longs;eguirebbe con &longs;imile modo il pe&longs;o po&longs;to in K e&longs;&longs;ere piu graue, che in I, &<emph.end type="italics"/> <pb/><emph type="italics"/>mouer&longs;i &longs;olamente fin ad S. </s> <s>Et &longs;e di nouo mo&longs;tra&longs;&longs;ero vna portione della &longs;ce&longs;a da S <lb/>in A, & co&longs;i &longs;ucceßiuamente e&longs;&longs;ere piu diritta della &longs;ce&longs;a eguale del pe&longs;o oppo&longs;to; <lb/>&longs;empre &longs;eguirà, che la bilancia SI andarà piu da pre&longs;&longs;o ad AB, ma non <expan abbr="dimostre-rãno">dimostre­<lb/>ranno</expan> giamai che per <lb/>uenga in AB. </s> <s>Se <lb/>dunque vogliono di <lb/>mo&longs;trare, che la <expan abbr="bilã">bilam</expan> <lb/>cia DE ritorni in <lb/>AB, egli è nece&longs;&longs;a­<lb/>rio, che pre&longs;upponga <lb/>no, che la &longs;ce&longs;a del <lb/>pe&longs;o da D in A <expan abbr="pr&etilde;">prem</expan> <lb/>da di diretto la quan <lb/>tità della linea tira­<lb/>ta dal punto D ad <lb/>AB ad angoli ret­<lb/>ti; & co&longs;i, &longs;e para­<lb/>goneremo le &longs;ce&longs;e e­<lb/>guali di DA AN <lb/>fra loro, lequali <expan abbr="pr&etilde;">prem</expan> <lb/>dono di diretto OC <lb/>CT, accaderà, che<emph.end type="italics"/><lb/> <arrow.to.target n="fig18"></arrow.to.target><lb/><emph type="italics"/>il pe&longs;o i&longs;te&longs;&longs;o &longs;arà in D graue egualmente, come in A. </s> <s>Ma &longs;e le portioni &longs;olamente <lb/>piglieremo da DA, &longs;arà piu graue in A, che in D. </s> <s>Adunque dalla diuer&longs;ità &longs;o­<lb/>lamente del modo del con&longs;iderare, auerrà, che il pe&longs;o mede&longs;imo &longs;arà & piu graue, <lb/>& piu leggiero; & non per la natura della co&longs;a. </s> <s>Di piu la pre&longs;uppo&longs;ta loro non <lb/>afferma, che il pe&longs;o &longs;econdo il &longs;ito &longs;ia piu graue, quanto nel &longs;ito mede&longs;imo il principio <lb/>della &longs;ua di&longs;ce&longs;a è meno obliquo. </s> <s>La pre&longs;upposta dunque di &longs;opra addotta, cioè che <lb/>&longs;econdo il &longs;ito il pe&longs;o è piu graue quanto nell'i&longs;te&longs;&longs;o &longs;ito meno obliqua è la di&longs;ce&longs;a, non <lb/>&longs;olamente non &longs;i puote concedere à modo alcuno, per le co&longs;e, che habbiamo dette; <lb/>ma anco percioche non è co&longs;a difficile il dimo&longs;trare tutto l'oppo&longs;to, cioè il pe&longs;o mede&longs;i <lb/>mo in eguali circonferenze quanto meno obliqua è la di&longs;ce&longs;a, iui meno grauare.<emph.end type="italics"/></s> </p> <figure id="fig18"></figure> <p type="main"> <s><emph type="italics"/>Siano come prima le circonferenze AL AM tra loro eguali; & &longs;ia il punto L vici <lb/>no ad F, & congiunga&longs;i LM, la quale &longs;arà à piombo di AB & LX &longs;arà anco <lb/>eguale ad XM. </s> <s>Dapoi pre&longs;&longs;o ad M tra M & G &longs;ia pre&longs;o come &longs;i vuole, il pun<emph.end type="italics"/><lb/> <arrow.to.target n="note48"></arrow.to.target> <emph type="italics"/>to P, & &longs;ia fatta la circonferenza PO eguale alla circonferenza AM, &longs;arà il <lb/>punto O pre&longs;&longs;o ad A. </s> <s>& &longs;iano congiunte le linee CL, CO, CM, CP, OP.<emph.end type="italics"/><lb/> <arrow.to.target n="note49"></arrow.to.target> <emph type="italics"/>& dal punto P tiri&longs;i la PN a piombo di OC. </s> <s>& percioche la circonferenza. </s> <s><lb/>AM è eguale alla circonferentia OP; &longs;arà l'angolo ACM eguale all'angolo<emph.end type="italics"/><lb/> <arrow.to.target n="note50"></arrow.to.target> <emph type="italics"/>OCP, & l'angolo CXM retto eguale al retto CNP, &longs;arà anco il re&longs;tante angolo <lb/>XMC del triangolo MXC eguale al re&longs;tante NPC del triangolo PCN.<emph.end type="italics"/> <pb pagenum="16"/><emph type="italics"/>Ma il lato ancora CM è eguale al lato CP, dunque il triangolo MCX è egua <lb/>le al triangolo PCN, & il lato MX eguale al lato NP. </s> <s>Onde la linea PN <lb/>&longs;arà eguale ad LX. </s> <s>Tiri&longs;i oltre a ciò dal punto O la linea OT egualmente di­<lb/>&longs;tante da AC, laquale tagli NP in V. & &longs;ia anco tirata dal punto P vna <lb/>linea a piombo di OT, <lb/>la quale per certo non <lb/>puote cadere tra OV, <lb/>perche e&longs;&longs;endo l'angolo <lb/>ONV retto, &longs;arà acu<emph.end type="italics"/> <arrow.to.target n="note51"></arrow.to.target><lb/><emph type="italics"/>to lo OVN. </s> <s>Per la <lb/>qualco&longs;a OVP &longs;arà <lb/>ottu&longs;o. </s> <s>Non caderà <lb/>dunque la linea tirata <lb/>dal punto P tra OV <lb/>à piombo di OT: pe­<lb/>roche due angoli d'uno <lb/><expan abbr="triãgolo">triangolo</expan> &longs;arebbono l'u­<lb/>no retto, & l'altro ot­<lb/>tu&longs;o, che è impoßibile. </s> <s><lb/>Caderà dun que nella li <lb/>nea OT nellaparte di <lb/>VT, et &longs;ia PT. </s> <s>&longs;arà &longs;e <lb/>condo e&longs;&longs;i, PT la di<emph.end type="italics"/><lb/> <arrow.to.target n="fig19"></arrow.to.target><lb/><emph type="italics"/>ritta &longs;ce&longs;a della circonferenza OP. </s> <s>Percioche dunque l'angolo ONV è retto,<emph.end type="italics"/> <arrow.to.target n="note52"></arrow.to.target><lb/><emph type="italics"/>&longs;arà la linea OV maggiore della ON. </s> <s>Onde la OT &longs;arà parimente maggiore <lb/>della ON. </s> <s>& co&longs;i di&longs;tendendo&longs;i la linea OP &longs;otto gli angoli retti ONP, <lb/>OTP, &longs;arà il quadrato di OP eguale alli quadrati ON NP in&longs;ieme pre&longs;i, &longs;i<emph.end type="italics"/> <arrow.to.target n="note53"></arrow.to.target><lb/><emph type="italics"/>milmente eguale a i quadrati di OT TP in&longs;ieme. </s> <s>per laqual co&longs;a li quadrati in&longs;ie­<lb/>me di ON NP &longs;aranno eguali a i quadrati in&longs;ieme di OT TP. </s> <s>Ma il quadrato <lb/>di OT è maggiore del quadrato di ON. per e&longs;&longs;ere maggiore la linea OT della <lb/>ON. </s> <s>Adunque il quadrato di NP &longs;ara maggiore del quadrato TP & perciò la <lb/>linea TP &longs;arà minore della linea PN, & della linea LX. </s> <s>Meno obliqua <lb/>dunque &longs;arà la &longs;ce&longs;a dell'arco LA, che dell'arco OP. </s> <s>Dunque il pe&longs;o po­<lb/>sto in L, per i loro detti, &longs;arà piu graue, che in O, il che, per le co&longs;e, che di <lb/>&longs;opra habbiamo detto, è manife&longs;tamente fal&longs;o. </s> <s>concio&longs;ia, che il pe&longs;o po&longs;to in O <lb/>&longs;ia piu graue, che in L. </s> <s>Non &longs;i puote dunque raccogliere dal piu diritto, & <lb/>piu torto mouimento in quel modo pigliato, e&longs;&longs;ere il pe&longs;o tanto piu graue &longs;econ­<lb/>do il &longs;ito, quanto nel mede&longs;imo &longs;ito è meno torta la &longs;ce&longs;a. </s> <s>& quinci na&longs;ce tutto <lb/>qua&longs;i il &longs;uo errore & inganno in cote&longs;ta co&longs;a. </s> <s>Imperoche quantunque per acciden­<lb/>te alle volte dalle co&longs;e fal&longs;e ne &longs;egua il vero, tutta via per &longs;e &longs;te&longs;&longs;e principalmente <lb/>dalle fal&longs;e ne &longs;egue il fal&longs;o, &longs;i come dalle vere &longs;empre il vero ne &longs;egue. </s> <s>Non è pero <lb/>da mar auigliar&longs;i, &longs;e mentre e&longs;&longs;i prendono co&longs;e fal&longs;e, & &longs;tanno &longs;opra quelle, come ve<emph.end type="italics"/> <pb/><emph type="italics"/>ri&longs;&longs;ime, raccolgono, & conchiudono co&longs;e in tutto fal&longs;i&longs;&longs;ime. </s> <s>&longs;ono oltre a ciò inganna­<lb/>ti, mentre pigliano a contemplare la bilancia &longs;emplicemente per via di matematica, <lb/>e&longs;&longs;endo la con&longs;ideratione &longs;ua mechanica affatto, ne di lei &longs;i po&longs;&longs;a ragionare a modo al <lb/>cuno &longs;enza il vero mouimento, & &longs;enza i pe&longs;i, che &longs;ono in tutto co&longs;e naturali, &longs;en­<lb/>za le quali non &longs;i po&longs;&longs;ono ritrouare per niuna maniera le vere cagioni di quelle co&longs;e, <lb/>che accadono alla bilancia.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note48"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 27. <emph type="italics"/>del terzo<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note49"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 32. <emph type="italics"/>del primo<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note50"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 26. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note51"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 13. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note52"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 19. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note53"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 47. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure id="fig19"></figure> <p type="main"> <s><emph type="italics"/>Oltre a ciò &longs;e anche con <lb/>cederemo la pre&longs;up­<lb/>po&longs;ta, &longs;i partono tut <lb/>tauia molto <expan abbr="lũge">lunge</expan> dal <lb/>la <expan abbr="cõ&longs;ideratione">con&longs;ideratione</expan> della <lb/>bilancia, mentre di­<lb/>&longs;corrono; che in quel <lb/>la maniera debba la <lb/>bilancia DE ritor­<lb/>nare in AB: percio <lb/>che &longs;empre pigliano <lb/>vn di due pe&longs;i &longs;epara <lb/>tamente come D, <lb/>ouero E, come &longs;e hor <lb/>l'uno, hor l'altro fo&longs; <lb/>&longs;e po&longs;to nella bilan­<lb/>cia, non congiunti in <lb/>&longs;ieme ambidue in <lb/>modo veruno, e&longs;&longs;en-<emph.end type="italics"/><lb/> <arrow.to.target n="fig20"></arrow.to.target><lb/><emph type="italics"/>doche nondimeno bi&longs;ogni fare tutto all'oppo&longs;ito di ciò, ne &longs;i puote con&longs;iderare dirit­<lb/>tamente l'uno &longs;enza l'altro, e&longs;&longs;endoche &longs;i ragiona di loro nella bilancia collocati. </s> <s><lb/>Concio&longs;ia che quando dicono la di&longs;ce&longs;a del pe&longs;o po&longs;to in D e&longs;&longs;ere meno torta, che <lb/>la di&longs;ce&longs;a del pe&longs;o po&longs;to in E, co&longs;i &longs;arà il pe&longs;o in D, per la pre&longs;uppo&longs;ta, piu graue <lb/>del pe&longs;o po&longs;to in E; onde per e&longs;&longs;ere piu graue, eglie nece&longs;&longs;ario, che &longs;i moua in giu, <lb/>& che la bilancia DE ritorni in AB: Cote&longs;to di&longs;cor&longs;o non è di momento alcu­<lb/>no. </s> <s>Primieramente &longs;empre argomentano come &longs;e i pe&longs;i in DE debbano &longs;cende­<lb/>re, con&longs;iderando la &longs;ce&longs;a di vno &longs;olameute &longs;enza la compagnia, & congiungimen­<lb/>to dell'altro. </s> <s>Vltimamente nondimeno e&longs;&longs;i per la comparatione delle di&longs;ce&longs;e de'pe­<lb/>&longs;i conchiudono il pe&longs;o posto in D mouer&longs;i in giu, & il po&longs;to in E in &longs;u, prenden­<lb/>do l'uno, & l'altro pe&longs;o congiunti in&longs;ieme fra loro nella bilancia. </s> <s>Ma da &longs;uoi me­<lb/>de&longs;imi principij, i quali v&longs;ano, & dalle &longs;ue dimo&longs;trationi &longs;i puote cauare ageuoli&longs;&longs;i-<emph.end type="italics"/><lb/> <arrow.to.target n="note54"></arrow.to.target> <emph type="italics"/>mamente l'oppo&longs;ito di quel che &longs;i faticano di difendere. </s> <s>Imperoche &longs;e &longs;i paragona <lb/>la di&longs;ce&longs;a del pe&longs;o po&longs;to in D con la &longs;alita del pe&longs;o po&longs;to in E, come tirate le'linee <lb/>E<emph.end type="italics"/>K <emph type="italics"/>DH a piombo di AB, e&longs;&longs;endo l'angolo DCH eguale all'angolo ECK,<emph.end type="italics"/><lb/> <arrow.to.target n="note55"></arrow.to.target> <emph type="italics"/>& l'angolo DHC retto eguale al retto E<emph.end type="italics"/>K<emph type="italics"/>C, & il lato DC eguale al lato <lb/>CE; &longs;arà il triangolo CDH eguale al triangolo CEK, & il lato DH egua<emph.end type="italics"/> <pb pagenum="17"/><emph type="italics"/>le al lato EK: & e&longs;&longs;endo l'angolo DCA eguale all'angolo ECB, &longs;arà anco <lb/>la circonferenza DA eguale alla circonferenza BE. </s> <s>Mentre dunque il pe&longs;o po­<lb/>sto in D &longs;cende per la circonferenza DA, il pe&longs;o po&longs;to in E &longs;ale per la circon­<lb/>ferenza EB eguale a DA, & la &longs;ce&longs;a del pe&longs;o po&longs;to in D prenderà, (&longs;econdo <lb/>il co&longs;tume loro) di diretto DH: & la &longs;alita del pe&longs;o E prenderà di diretto EK <lb/>eguale a DH: &longs;arà dunque la &longs;ce&longs;a del pe&longs;o posto in D eguale alla &longs;alita del pe&longs;o <lb/>po&longs;to in E: & quale &longs;arà la inclinatione d'uno al mouimento in giù, tale &longs;arà etian <lb/>dio la re&longs;i&longs;tenza dell'altro al mouimento in sù, cioè la re&longs;istentia della violenza del <lb/>pe&longs;o po&longs;to in E nella a&longs;ce&longs;a, contra&longs;tando &longs;i oppone alla naturale po&longs;&longs;anza del pe­<lb/>&longs;o po&longs;to in D per e&longs;&longs;ere a lei eguale; percioche quanto il pe&longs;o po&longs;to in D per la na­<lb/>tural po&longs;&longs;anza de&longs;cende piu velocemente in giù, in tanto il pe&longs;o po&longs;to in E più tar­<lb/>do &longs;ale violentemente. </s> <s>Per laqual co&longs;a niuno di loro due pe&longs;era piu dell'altro, non <lb/>procedendo attione da eguale. </s> <s>il pe&longs;o po&longs;to in D dunque non mouerà il pe&longs;o po&longs;to <lb/>in E in &longs;u&longs;o, peroche &longs;e lo moue&longs;&longs;e, &longs;arebbe nece&longs;&longs;ario, che il pe&longs;o po&longs;to in D ha­<lb/>ue&longs;&longs;e virtu maggiore in di&longs;cendendo, che il pe&longs;o po&longs;to in E in &longs;alendo, ma que&longs;te co­<lb/>&longs;e &longs;ono eguali: adunque &longs;taranno &longs;ermi i pe&longs;i, & la grauezza del pe&longs;o po&longs;to in D &longs;a­<lb/>rà eguale alla grauezza del pe&longs;o po&longs;to in E. </s> <s>Oltre a ciò perche pre&longs;uppongono, che <lb/>quanto il pe&longs;o è piu di&longs;tante dalla linea FG della dirittura, tanto e&longs;&longs;ere piu graue. </s> <s><lb/>però tirate parimente da i punti DE le linee DO, EI apiombo di FG, con <lb/>modo &longs;imile &longs;i dimostrerà il triangolo CDO e&longs;&longs;ere eguale al triangolo CEI: & <lb/>la linea DO e&longs;&longs;ere eguale ad EI. </s> <s>Tanto dunque è di&longs;tante il pe&longs;o po&longs;to in D <lb/>dalla linea FG, quanto il pe&longs;o po&longs;to in E. </s> <s>Dalle ragioni loro dunque, & dalle &longs;ue <lb/>pre&longs;uppo&longs;te li pe&longs;i me&longs;&longs;i in DE &longs;ono graui egualmente. </s> <s>Di piu, che vieta che non &longs;i di <lb/>mo&longs;tri la bi lancia DE mouer&longs;i per nece&longs;&longs;ità in FG con &longs;imile ragione? </s> <s>Primie­<lb/>ramente &longs;i puote raccogliere dalle loro mede&longs;ime dimo&longs;trationi, la &longs;alita del pe&longs;o po­<lb/>&longs;to in E ver&longs;o il B e&longs;&longs;ere piu diritta della &longs;alita del pe&longs;o po&longs;to in D ver&longs;o lo F, <lb/>cioè manco prendere di diretto la &longs;alita del pe&longs;o po&longs;to in D in archi eguali, che la <lb/>&longs;alita del pe&longs;o po&longs;to in E. </s> <s>Pre&longs;upponga&longs;i dunque, che il pe&longs;o &longs;ia piu leggiero &longs;econ­<lb/>do il &longs;ito tanto quanto nel &longs;ito mede&longs;imo meno diritta è la &longs;ua &longs;alita: Laqual pre­<lb/>&longs;upposta pare tanto manife&longs;ta, quanto l'altraloro. </s> <s>percioche dunque la &longs;alita del <lb/>pe&longs;o po&longs;to in E è piu diritta della &longs;alita del pe&longs;o po&longs;to in D, per la pre&longs;uppo&longs;ta il <lb/>pe&longs;o po&longs;to in D &longs;arà piu leggiero del pe&longs;o po&longs;to in E. </s> <s>Adunque il pe&longs;o po&longs;to in D <lb/>&longs;i mouerà in sù dal pe&longs;o po&longs;to in E, &longs;i &longs;attamente che la bilancia peruenga in FG, <lb/>& co&longs;i potra&longs;si dimo&longs;trare la bilancia DE mouer&longs;i in FG, laqual dimo&longs;tratio­<lb/>ne è del tutto veramente friuola, & pati&longs;ce le difficultà mede&longs;ime. </s> <s>Percioche quan­<lb/>tunque &longs;i conceda, come vero, che il pe&longs;o po&longs;to in E &longs;alendo &longs;ia piu graue del pe&longs;o <lb/>in D &longs;imilmente &longs;alendo, non perciò da que&longs;to &longs;egue, che il pe&longs;o po&longs;to in E de­<lb/>&longs;cendendo &longs;ia piu graue del pe&longs;o posto in D &longs;alendo. </s> <s>Niuna dunque di que&longs;te due <lb/>dimo&longs;trationi, che dicono la bilancia DE ritornare in AB, ouero mouer&longs;i in <lb/>FG, è vera.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note54"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note55"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 25. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure id="fig20"></figure> <p type="main"> <s><emph type="italics"/>Oltre a ciò &longs;e e&longs;amineremo la loro pre&longs;uppo&longs;ta, & la &longs;orza delle loro parole, vedremo <lb/>per certo che altro &longs;entimento hanno. </s> <s>Imperoche e&longs;&longs;endo che &longs;empre lo &longs;patio per lo<emph.end type="italics"/> <pb/><emph type="italics"/>quale il pe&longs;o natur <expan abbr="alm&etilde;te">almente</expan> &longs;i moue, &longs;i deue prendere dal centro della grauezza di e&longs;­<lb/>&longs;o pe&longs;o ver&longs;o il centro del mondo à &longs;embianza di vna linea diritta tirata dal centro <lb/>della grauezza al centro del mondo, tanto &longs;i dir à que&longs;ta co&longs;i fatta di&longs;ce&longs;a del pe&longs;o <lb/>piu, & meno obliqua, quanto, &longs;econdo lo &longs;patio di&longs;&longs;egnato, a &longs;embianza della pre­<lb/>detta linea piu ò meno &longs;i mouerà, (andando pero &longs;empre a trouare il luogo &longs;uo natu <lb/>rale, & vie piu &longs;empre auicinandoui&longs;i.) talche tanto piu obliqua &longs;i dica la &longs;ce&longs;a <expan abbr="quã">quam</expan> <lb/>to &longs;i parte da cotale &longs;patio: & piu diritta quanto a lui &longs;i acco&longs;ta. </s> <s>& in que&longs;to <lb/>&longs;entimento quella pre&longs;upposta non deue partorire difficulta ad alcuno, percioche co­<lb/>&longs;i è la verita &longs;ua chiara, & conforme alla ragione, che non pare hauer me&longs;tieri di e&longs;­<lb/>&longs;er &longs;atta in alcun modo manife&longs;ta.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/><gap/>e dunque il pe&longs;o &longs;ciolto, collocato nel &longs;i­<lb/>to di D &longs;i deue mouere al luogo pro­<lb/>prio, &longs;enza dubbio, po&longs;to S centro del <lb/>mondo, &longs;i mouerà per la linea DS, <expan abbr="&longs;i-milm&etilde;te">&longs;i­<lb/>milmente</expan> il pe&longs;o po&longs;to in E &longs;ciolto &longs;i mo <lb/>uerà per la linea ES. </s> <s>Per laqual co­<lb/>&longs;a &longs;e, (come è vero) la &longs;ce&longs;a del pe&longs;o &longs;i <lb/>dirà piu, ò meno obliqua, &longs;econdo lo al <lb/>lontanar&longs;i, ouero appre&longs;&longs;ar&longs;i a gli &longs;patij <lb/>di&longs;segnati per le linee DS ES, per ri <lb/>&longs;petto a'loro naturali mouimenti ver&longs;o <lb/>iproprij luoghi, egli è chiaro, che meno <lb/>obliqua è la &longs;ce&longs;a di E per EG, che <lb/>di D per DA, per e&longs;&longs;ere stato di <lb/>&longs;opra mo&longs;trato che l'angolo SEG è <lb/>minore dell'angolo SDA. </s> <s>Per laqual <lb/>co&longs;a piu grauer à il pe&longs;o in E, che in D, <lb/>il che totalmente è il contrario di quel­<lb/>lo, che e&longs;si &longs;i &longs;ono s&longs;orzati di prouare. </s> <s><lb/>Leueran&longs;i per auuentura contra di noi <lb/>dicendo. </s> <s>Se dundue il pe&longs;o po&longs;to in E è <lb/>piu graue del pe&longs;o po&longs;to in D, la bi­<lb/>lancia DE non &longs;tar à giamai in que­<lb/>&longs;to &longs;ito, laqual co&longs;a noi habbiamo pro­<lb/>po&longs;to di mantenere, ma &longs;i mouer à in F <lb/>G. </s> <s>Allequali co&longs;e ri&longs;pondiamo. </s> <s>che im­<lb/>porta a&longs;&longs;ai, &longs;e noi con&longs;ideriamo i pe&longs;i o­<lb/>uero in quanto &longs;ono &longs;eparati l'uno dal­<lb/>l'altro, ouero in quanto &longs;ono traloro <lb/>congiunti: perche altra è la ragione del<emph.end type="italics"/><lb/> <arrow.to.target n="fig21"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o po&longs;to in E &longs;enza il congiungimento del pe&longs;o po&longs;to in D, & altra di lui con <lb/>l'altro pe&longs;o congiunto, &longs;i fattamente che l'uno &longs;enza l'altro non &longs;i po&longs;&longs;a mouere. </s> <s>Im<emph.end type="italics"/> <pb pagenum="18"/><emph type="italics"/>peroche la diritta, & naturale di&longs;ce&longs;a dal pe&longs;o po&longs;to in E, inquanto egli è &longs;enza al­<lb/>tro congiungimento di pe&longs;o, &longs;i fa per la linea ES. ma inquanto egli è congiunto <lb/>col pe&longs;o D, la &longs;ua naturale di&longs;ce&longs;a non &longs;arà piu per la linea ES, ma per vna li­<lb/>nea egualmente di&longs;tante da CS. </s> <s>percioche la magnitudine compo&longs;ta de i pe&longs;i ED. <lb/>& della bilancia DE il cui centro della grauezza è C, &longs;e in ne&longs;&longs;un luogo non &longs;a­<lb/>rà &longs;o&longs;tenuta, &longs;i mouerà naturalmente in giu nel modo che &longs;i troua, &longs;econdo la gra­<lb/>uezza del centro per la linea diritta tirata dal centro della grauezza C al centro <lb/>del mondo S, finche il centro C peruenga nel centro S. </s> <s>La bilancia dunque DE <lb/>in&longs;ieme co'pe&longs;i, in quella maniera, che &longs;i troua &longs;i mouerà in giu per modo tale, che il <lb/>punto C &longs;i moua per la linea CS, fin che C peruenga in S, & la bilancia <lb/>DE in HK; & habbia la bilancia in HK la po&longs;itione i&longs;te&longs;&longs;a, che prima hauea; <lb/>cio è, che la HK &longs;ia egualmente distante da DE. </s> <s>Congiungan&longs;i dunque DH <lb/>EK. </s> <s>egli è manife&longs;to, che mentre la bilancia DE &longs;i moue in HK, mouer&longs;i an­<lb/>che ipunti DE per le linee DH EK, come quelle che &longs;ono & &longs;ra &longs;e, & ad<emph.end type="italics"/> <arrow.to.target n="note56"></arrow.to.target><lb/><emph type="italics"/>e&longs;&longs;a CS eguali, & egualmente di&longs;tanti. </s> <s>Per la qual co&longs;a i pe&longs;i posti in DE, in <lb/>quanto &longs;ono &longs;ra loro congiunti, &longs;e riguarderemo il mouimento loro naturale &longs;imoue <lb/>ranno non &longs;econdo le linee DS, ES, ma &longs;econdo LDH MEK egualmente <lb/>di&longs;tanti da e&longs;&longs;a CS. </s> <s>Ma la naturale inclinatione del pe&longs;o po&longs;to in E libero, & <lb/>&longs;ciolto &longs;arà per ES, & del pe&longs;o po&longs;to in D <expan abbr="&longs;imilm&etilde;te">&longs;imilmente</expan> &longs;ciolto &longs;arà per DS. & per­<lb/>cio non è &longs;conueneuole, che il pe&longs;o mede&longs;imo hora in E, hora in D, &longs;ia piu graue <lb/>in E, che in D. </s> <s>Ma&longs;e i pe&longs;i po&longs;ti in ED &longs;ono l'un l'altro fra &longs;e congiunti, & gli <lb/>con&longs;idereremo in quanto &longs;ono congiunti, &longs;arà la naturale inclinatione del pe­<lb/>&longs;o po&longs;to in E per la linea MEK, percioche la grauezza dell'altro pe&longs;o po&longs;to <lb/>in D fa &longs;i, che il pe&longs;o po&longs;to in E non graui &longs;opra la linea ES, ma nella EK. </s> <s><lb/>Ilche fa parimente la grauezza del pe&longs;o po&longs;to in E, cioè, che il pe&longs;o po&longs;to in D <lb/>non graui per la linea retta DS, ma &longs;econdo DH, per impedir&longs;i ambedue l'uno <lb/>l'altro che non vadino à propri luoghi. </s> <s>Concio&longs;ia dunque che la naturale &longs;ce&longs;a dirit­<lb/>ta de i pe&longs;i po&longs;ti in DE &longs;ia &longs;econdo LDH, MEK, &longs;arà &longs;imilmente la naturale <lb/>&longs;alita diritta loro &longs;econdo le i&longs;te&longs;&longs;e linee HDL KEM. & la naturale &longs;alita del <lb/>pe&longs;o po&longs;to in E &longs;i dirà più, & meno torta, quanto che &longs;econdo lo &longs;patio &longs;i mouerà <lb/>più, & meno pre&longs;&longs;o la linea MK. </s> <s>& a que&longs;to modo in tutto &longs;i ha da pigliare & la &longs;a <lb/>lita & la di&longs;ce&longs;a del pe&longs;o po&longs;to in D &longs;econdo la linea LH, &longs;e dunque il pe&longs;o po&longs;to <lb/>in E &longs;i moue&longs;&longs;e in giù per la linea EG, mouerebbe il pe&longs;o po&longs;to in D in sù per <lb/>DF. & percioche l'angolo CEK è eguale all'angolo CDL, & l'angolo CEG<emph.end type="italics"/> <arrow.to.target n="note57"></arrow.to.target><lb/><emph type="italics"/>è eguale all'angolo CDF; &longs;arà il <expan abbr="re&longs;tãte">re&longs;tante</expan> angolo GEK al re&longs;tante LDF egua <lb/>le. </s> <s>& e&longs;&longs;endo quella pre&longs;uppo&longs;ta, che dice il pe&longs;o e&longs;&longs;er più graue &longs;econdo il &longs;ito, <lb/>quanto in quel mede&longs;imo &longs;ito la di&longs;ce&longs;a è meno obliqua per chiara, & manife&longs;ta ri­<lb/>ceuuta, &longs;arà anche da e&longs;&longs;ere accettata &longs;enza dubbio que&longs;t' altra, cioè, che il pe&longs;o &longs;arà <lb/>più graue &longs;econdo il &longs;ito, quanto nel &longs;ito mede&longs;imo meno obliqua &longs;arà la &longs;alita; per <lb/>non e&longs;&longs;ere manco manife&longs;ta, ne meno conforme alla ragione. </s> <s>&longs;arà dunque eguale <lb/>la &longs;ce&longs;a del pe&longs;o po&longs;to in E alla &longs;alita del pe&longs;o po&longs;to in D, percioche la &longs;ce&longs;a del pe <lb/>&longs;o po&longs;to in E tiene tanto di obliquo, quanto la &longs;alita del pe&longs;o po&longs;to in D. & quale<emph.end type="italics"/></s> </p> <pb/> <p type="margin"> <s><margin.target id="note56"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 35. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note57"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure id="fig21"></figure> <p type="head"> <s>PROPOSITIONE V.</s> </p> <p type="main"> <s>Due pe&longs;i attaccati nella bilancia, &longs;e la bilancia &longs;arà tra loro in modo <lb/>diui&longs;a, chele parti ri&longs;pondano &longs;cambieuolmente à pe&longs;i; pe&longs;eranno <lb/>tanto ne'punti doue &longs;ono attaccati, quanto &longs;el'uno & l'altro fo&longs;&longs;e <lb/>pendente dal punto della diui&longs;ione. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Siala bilancia AB, il cui centro &longs;ia C, & &longs;iano due pe&longs;i EF pendenti da' punti <lb/>BG: & diuida&longs;i BG in H, &longs;i fattamente, che BH ad HG habbia la pro­<lb/>portione iste&longs;&longs;a, che hà il pe&longs;o E al pe&longs;o F. </s> <s>Dico ipe&longs;i EF pe&longs;are tanto in BG, <lb/>quanto &longs;e amendue pende&longs;&longs;ero dal punto H. </s> <s>faccia&longs;i AC eguale à CH. & &longs;i <lb/>come AC à CG, co&longs;i &longs;accia&longs;i il pe&longs;o E al pe&longs;o L. &longs;imilmente come AC à <lb/>CB, co&longs;i faccia&longs;i il pe&longs;o F al pe&longs;o M. & &longs;iano attaccati ipe&longs;i LM al punto <lb/>A. </s> <s>Horpercioche AC è eguale à CH, &longs;arà BC ver&longs;o CH come il pe&longs;o <lb/>M alpe&longs;o F. & percioche piu grande è BC di CH; &longs;arà anche il pe&longs;o M<emph.end type="italics"/><lb/> <arrow.to.target n="note58"></arrow.to.target> <emph type="italics"/>maggiore di F. </s> <s>Diuida&longs;i dunque il pe&longs;o M in due parti QR, & &longs;iala parte di<emph.end type="italics"/><lb/> <arrow.to.target n="note59"></arrow.to.target> <emph type="italics"/>Q eguale ad F; &longs;arà BC à CH, come RQ à Q: & diuidendo, come BH <lb/>ad HC, co&longs;i R à <expan abbr="q.">que</expan> Dapoi conuertendo, come CH ad HB, co&longs;i Q ad<emph.end type="italics"/><lb/> <arrow.to.target n="note60"></arrow.to.target> <emph type="italics"/>R. </s> <s>Oltre à ciò perche CH è eguale à CA, &longs;arà HC ver&longs;o CG come il pe&longs;o <lb/>E al pe&longs;o L: maè piu grande HC di CG, però &longs;arà anche il pe&longs;o E maggio-<emph.end type="italics"/><lb/> <arrow.to.target n="note61"></arrow.to.target> <emph type="italics"/>re del pe&longs;o L. </s> <s>Onde diuida&longs;i il pe&longs;o E in due parti NO, &longs;i fattamente, che la <lb/>parte di O &longs;ia eguale ad L, &longs;arà HC à CG come tutto lo NO ad O; & <lb/>diuidendo, come HG à GC, co&longs;i N ad O. & conuertendo, come CG à<emph.end type="italics"/><lb/> <arrow.to.target n="note62"></arrow.to.target> <emph type="italics"/>GH, co&longs;i O ad N. & di nuouo componendo, come CH ad HG, co&longs;i ON<emph.end type="italics"/><lb/> <arrow.to.target n="note63"></arrow.to.target> <emph type="italics"/>ad N. </s> <s>& come GH ad HB, co&longs;i è F ad ON. </s> <s>Per la qual co&longs;a per la pro <lb/>portione vguale come CH ad HB, co&longs;i F ad N. </s> <s>Ma come CH ad HB<emph.end type="italics"/><lb/> <arrow.to.target n="note64"></arrow.to.target> <emph type="italics"/>co&longs;i è Q ad R: &longs;arà dunque Q ad R come F ad N. & permutando co­<lb/>me Q ad F; co&longs;i R ad N. </s> <s>ma la parte di Q è egual ad e&longs;&longs;o F. per la qual<emph.end type="italics"/><lb/> <arrow.to.target n="note65"></arrow.to.target> <emph type="italics"/>co&longs;a la parte di R ancora &longs;arà eguale ad N. </s> <s>e&longs;&longs;endo dunque il pe&longs;o L eguale<emph.end type="italics"/> <pb pagenum="30"/><emph type="italics"/>ad O, & il pe&longs;o F eguale parimente al Q, & la parte di R eguale ad N; &longs;a <lb/>ranno ipe&longs;i LM eguali a i pe&longs;i E\1</s> <s>\2amp; percioche &longs;i come AC ver&longs;o CG, co<emph.end type="italics"/> <arrow.to.target n="note66"></arrow.to.target><lb/><emph type="italics"/>&longs;i è il pe&longs;o E al pe&longs;o L, ipe&longs;i EL pe&longs;eranno egualmente. </s> <s>&longs;imilmente percioche <lb/>&longs;i come AC è ver&longs;o CB, co&longs;i il pe&longs;o F è alpe&longs;o M, i pe&longs;i FM pe&longs;eranno <lb/>anco egualmente. </s> <s>i pe&longs;i dunque LM pe&longs;eranuo egualmente co'pe&longs;i EF attacca­<lb/>ti in BG. </s> <s>& e&longs;&longs;endo la di&longs;tanza CA eguale alla di&longs;tanza CH, &longs;e dunque am <lb/>bidue i pe&longs;i EF &longs;ar anno attaccati in H, i pe&longs;i LM pe&longs;er anno egualmente co'<emph.end type="italics"/> <arrow.to.target n="note67"></arrow.to.target><lb/><emph type="italics"/>pe&longs;i EF attaccati in H. </s> <s>Ma LM pe&longs;a ancora egualmente con EF in GB. </s> <s><lb/>Adunque &longs;ar anno egualmente graui i pe&longs;i EF in GB attaccati come in H. </s> <s>pe<emph.end type="italics"/> <arrow.to.target n="note68"></arrow.to.target><lb/><emph type="italics"/>&longs;eranno dunque tanto in BG quanto attaccati in H.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note58"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note59"></margin.target><emph type="italics"/>Per la con-<expan abbr="&longs;egu&etilde;za">&longs;eguenza</expan> del la<emph.end type="italics"/> 4. <emph type="italics"/>del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note60"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note61"></margin.target><emph type="italics"/>Per la con&longs;e <expan abbr="gu&etilde;za">guenza</expan> della<emph.end type="italics"/> 4. <emph type="italics"/>del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note62"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note63"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note64"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note65"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note66"></margin.target><emph type="italics"/>Perla<emph.end type="italics"/> 6. <emph type="italics"/>del primo di<gap/>r chimede del le co&longs;e che pe &longs;ano egualmente.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note67"></margin.target><emph type="italics"/>Per lo<emph.end type="italics"/> 2. <emph type="italics"/><expan abbr="cõ">com</expan>. </s> <s>della not di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note68"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/><expan abbr="cõ">com</expan>. </s> <s>della not ai questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;iano i pe&longs;i EF attaccati in CB; & &longs;ia C il centro della bilancia, & diuida&longs;i <lb/>CB in H, per modo che CH ver&longs;o HB &longs;ia come il pe&longs;o F al pe&longs;o E. </s> <s>Dico <lb/>che i pe&longs;i EF pe&longs;er anno tanto in CB quanto nel punto H. </s> <s>faccia&longs;i CA egua <lb/>le à CH, & come CA ver&longs;o CB; co&longs;i faccia&longs;i il pe&longs;o F ver&longs;o vn'altro, che <lb/>&longs;ia D, ilquale &longs;i appicchi in A. </s> <s>Hor percioche CH è eguale à CA, &longs;arà CH <lb/>ver&longs;o CB, come F à D; & ben è maggiore CB di CH, però il pe&longs;o D &longs;a <lb/>rà maggiore del pe&longs;o F. </s> <s>Diuida&longs;i dunque il D in due parti GK, & &longs;ia il G<emph.end type="italics"/> <arrow.to.target n="note69"></arrow.to.target><lb/><emph type="italics"/>eguale allo F; &longs;arà BC à CH come GK ver&longs;o il G; et diuidendo, come BH <lb/>ad HC, co&longs;i K ver&longs;o G; & conuertendo come CH ad HB, co&longs;i G ver-<emph.end type="italics"/> <arrow.to.target n="note70"></arrow.to.target><lb/><emph type="italics"/>&longs;o K. </s> <s>& come CH ad HB, co&longs;i è F ver&longs;o E. </s> <s>Dunque come G ver­<lb/>&longs;o K co&longs;i è F ad E. & permutando come G ad F, co&longs;i K ad E. & per-<emph.end type="italics"/> <arrow.to.target n="note71"></arrow.to.target><lb/><emph type="italics"/>che GF &longs;ono eguali, &longs;aranno anche KE tra loro eguali. </s> <s>Concio&longs;ia dunque che <lb/>la parte G &longs;ia eguale ad F, & il K ad e&longs;&longs;o E; &longs;arà tutto il GK eguale a i pe<emph.end type="italics"/> <arrow.to.target n="note72"></arrow.to.target><lb/><emph type="italics"/>&longs;i EF. & percioche AC è eguale à CH; &longs;e dunque i pe&longs;i EF &longs;aranno penden <lb/>ti dal punto H, il pe&longs;o D pe&longs;erà egualmente co'pe&longs;i EF attaccati in H. </s> <s>Ma <lb/>pe&longs;a anche egualmente con eßi in CB, cioè F in B, & E in C; per e&longs;&longs;ere <lb/>come AC ver&longs;o CB, co&longs;i F ver&longs;o D: percioche il pe&longs;o E pendente da C. <lb/>centro della bilancia non è cau&longs;a, che la bilancia &longs;i moua in alcuna delle due parti. </s> <s><lb/>tanto &longs;aranno dunque graui i pe&longs;i EF in CB, quanto in H appicati.<emph.end type="italics"/></s> </p> <pb/> <p type="margin"> <s><margin.target id="note69"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note70"></margin.target><emph type="italics"/>Per la con&longs;e <expan abbr="gu&etilde;za">guenza</expan> della<emph.end type="italics"/> 4. <emph type="italics"/>del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note71"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note72"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia <expan abbr="finalm&etilde;te">finalmente</expan> la <expan abbr="bilãcia">bilancia</expan> AB, & da i <expan abbr="pũti">punti</expan> AB &longs;iano <expan abbr="p&etilde;denti">pendenti</expan> ipe&longs;i EF, & &longs;idil centro <lb/>della bilancia C fra ipe&longs;i, & diuida&longs;i la AB in D, talche AD ver&longs;o DB <lb/>&longs;ia come il pe&longs;o F alpe&longs;o E. </s> <s>Dico che i pe&longs;i EF pe&longs;ano tanto in AB, quan <lb/>to &longs;e ambidue &longs;o&longs;&longs;ero pendenti dal punto D. </s> <s>faccia&longs;i CG eguale à CD; & co­<lb/>me DC à CA, co&longs;i faccia&longs;i il pe&longs;o E ad vn'altro pe&longs;o H, ilquale &longs;ia attac <lb/>cato in D. & come GC ver&longs;o CB, co&longs;i faccia&longs;i il pe&longs;o F ad vn'altro che <lb/>&longs;ia K, & attachi&longs;i K in G. </s> <s>Horpercioche, come il BC è ver&longs;o il CG, cioè <lb/>ver&longs;o il CD, co&longs;i il pe&longs;o K ad F; &longs;arà il K maggiore del pe&longs;o F. </s> <s>Per laqual <lb/>co&longs;a diuida&longs;i il pe&longs;o K in L & in MN, & &longs;accia&longs;i la parte L eguale ad F,<emph.end type="italics"/><lb/> <arrow.to.target n="note73"></arrow.to.target> <emph type="italics"/>&longs;arà come BC à CD, co&longs;i tutto LMN ad L; & diuidendo, come BD<emph.end type="italics"/><lb/> <arrow.to.target n="note74"></arrow.to.target> <emph type="italics"/>ver&longs;o DC, co&longs;i la parte MN alla parte L. come dunque BD à DC, co&longs;i <lb/>la parte MN ad F. & come AD à DB, co&longs;i F ad E. </s> <s>Per laqual co&longs;a<emph.end type="italics"/><lb/> <arrow.to.target n="note75"></arrow.to.target> <emph type="italics"/>per la egual proportione, come AD ver&longs;o DC, co&longs;i MN ad E. & e&longs;&longs;endo AD<emph.end type="italics"/><lb/> <arrow.to.target n="note76"></arrow.to.target> <emph type="italics"/>maggiore di CD; &longs;arà anco la parte MN maggiore del pe&longs;o E. </s> <s>Diuida&longs;i dun <lb/>que MN in due parti MN, & &longs;ia M eguale ad E. </s> <s>&longs;arà come AD à<emph.end type="italics"/><lb/> <arrow.to.target n="note77"></arrow.to.target> <emph type="italics"/>DC, co&longs;i NM ad M; & diuidendo, come AC ver&longs;o CD, co&longs;i N ad M:<emph.end type="italics"/><lb/> <arrow.to.target n="note78"></arrow.to.target> <emph type="italics"/>& conuertendo, come DC ver&longs;o CA, co&longs;i M ad N. </s> <s>& come DC à<emph.end type="italics"/><lb/> <arrow.to.target n="note79"></arrow.to.target> <emph type="italics"/>CA, co&longs;i è E ad H; &longs;arà dunque M ad N come E ad H; & permutan <lb/>do come M ad E, co&longs;i N ad H. </s> <s>Ma per e&longs;&longs;ere ME traloro eguali, &longs;aran­<lb/>no anche NH tra &longs;e eguali. </s> <s>& percioche co&longs;i è AC ver&longs;o CD, come H <lb/>ad E: i pe&longs;i HE pe&longs;eranno egualmente. </s> <s>&longs;imilmente percioche, come è GC à CB, <lb/>co&longs;i F ver&longs;o K, i pe&longs;i etiandio KF pe&longs;eranno egualmente. </s> <s>Adunque i pe&longs;i<emph.end type="italics"/><lb/> <arrow.to.target n="note80"></arrow.to.target> <emph type="italics"/>EK HF nella bilancia AB, il cui centro &longs;ia C pe&longs;eranno egualmente. </s> <s>& con <lb/>cio&longs;ia che GC &longs;ia eguale à CD, & il pe&longs;o H &longs;ia pur eguale ad N, ipe&longs;i NH<emph.end type="italics"/> <pb pagenum="31"/><emph type="italics"/>pe&longs;eranno egualmente. </s> <s>& percioche tutti pe&longs;ano egualmente, tolti via i pe&longs;i HN, <lb/>iquali pe&longs;ano egualmente, i re&longs;tanti pe&longs;eranno egualmente; cioè i pe&longs;i EF, & il pe<emph.end type="italics"/> <arrow.to.target n="note81"></arrow.to.target><lb/><emph type="italics"/>&longs;o LM pendenti dal centro C della bilancia. </s> <s>Ma percioche la parte L è egua­<lb/>le ad F, & la parte M è eguale alla parte E; &longs;arà tutto LM eguale a i pe&longs;i <lb/>FE in&longs;ieme pre&longs;i. </s> <s>& e&longs;&longs;endo CG eguale à CD, &longs;e i pe&longs;i EF &longs;aranno &longs;atti <lb/>pendenti dal punto D, ipe&longs;i EF appiccati in D pe&longs;eranno <expan abbr="egualm&etilde;te">egualmente</expan> con LM. </s> <s><lb/>Per laqual co&longs;a LM pe&longs;erà <expan abbr="egualmēte">egualmente</expan> <expan abbr="tāto">tanto</expan> ad eßi EF appiccati in AB, <expan abbr="quā-to">quan­<lb/>to</expan> &longs;e fo&longs;&longs;ero appiccati nel punto D; peroche la bilancia rimane &longs;empre nell'i&longs;te&longs;&longs;o<emph.end type="italics"/> <arrow.to.target n="note82"></arrow.to.target><lb/><emph type="italics"/>modo. </s> <s>Adunque i pe&longs;i EF pe&longs;eranno tanto in AB quanto nel punto D; che <lb/>bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note73"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note74"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 23. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note75"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note76"></margin.target><emph type="italics"/>Corollario della quarta del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note77"></margin.target>11. <emph type="italics"/>del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note78"></margin.target>16. <emph type="italics"/>del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note79"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del<emph.end type="italics"/> 1. <emph type="italics"/>di Archi mede delle co fa che egual <expan abbr="m&etilde;te">mente</expan> pe&longs;ano.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note80"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>no titia commu ne di que&longs;io.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note81"></margin.target><emph type="italics"/>Per la commune notitia di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note82"></margin.target><emph type="italics"/>Per la com munenotitia di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s>Ma que&longs;te co&longs;e tutte dimo&longs;treremo in altra maniera, & piu Mechani <lb/>camente. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia la bilancia AB, & il &longs;uo centro C, & &longs;iano, come nel primo ca&longs;o, due pe&longs;i EF <lb/>pendenti da i punti BG: & &longs;ia GH ad HB, come il pe&longs;o F al pe&longs;o E. </s> <s>Di­<lb/>co che ipe&longs;i EF pe&longs;eranno tanto in GB, quanto &longs;e ambidue &longs;te&longs;&longs;ero pendenti <lb/>dal punto H della diui&longs;ione. </s> <s>Siano di&longs;po&longs;te le mede&longs;ime co&longs;e, cioè faccia&longs;i AC <lb/>eguale à CH, & dal punto A &longs;iano appe&longs;i due pe&longs;i LM, per modo che il pe <lb/>&longs;o E ver&longs;o il pe&longs;o L &longs;ia come CA ver&longs;o CG; & come CB ver&longs;o CA, co <lb/>&longs;i &longs;ia il pe&longs;o M ver&longs;o il pe&longs;o F. </s> <s>Ipe&longs;i LM pe&longs;eranno egualmente (come è detto <lb/>di &longs;opra) conlipe&longs;i EF appiccati in GB. </s> <s>Siano dapoi due punti NO li centri <lb/>della grauezza de' pe&longs;i EF; & &longs;iano congiunte le linee GN BO; & &longs;ia con­<lb/>giunta NO, laquale &longs;arà come bilancia; laquale etiandio faccia sì, che le linee <lb/>GN BO &longs;iano traloro egualmente di&longs;tanti; & dal punto H &longs;ia tirata la HP <lb/>à piombo dell'orizonte, laquale tagli NO nel P, & &longs;ia egualmente distante dal <lb/>le linee GN BO. </s> <s>In fine congiunga&longs;i GO, laquale tagli HP in R. </s> <s>Percio<emph.end type="italics"/> <arrow.to.target n="note83"></arrow.to.target><lb/><emph type="italics"/>che dunque HR è egualmente di&longs;tante dal lato BO del triangolo GBO; &longs;arà <lb/>la GH ver&longs;ola HB, come GR ad RO. </s> <s>Similmente percioche RP è egual<emph.end type="italics"/> <pb/> <arrow.to.target n="fig22"></arrow.to.target><lb/><emph type="italics"/>mente di&longs;tante dal lato GN del triangolo OGN; &longs;arà GR ver&longs;o RO, come <lb/>NP ver&longs;o PO. </s> <s>Per laqual co&longs;a come GH ad HB, così è NP ver&longs;o PO.<emph.end type="italics"/><lb/> <arrow.to.target n="note84"></arrow.to.target> <emph type="italics"/>Ma come GH ver&longs;o HB, così è il pe&longs;o F ver&longs;o il pe&longs;o E; adunque come NP <lb/>ver&longs;o PO, così è il pe&longs;o F ver&longs;o il pe&longs;o E. </s> <s>Dunque il punto P &longs;arà il centro<emph.end type="italics"/><lb/> <arrow.to.target n="note85"></arrow.to.target> <emph type="italics"/>della grauezza della magnitudine compo&longs;ta di ambidue i pe&longs;i EF. </s> <s>Intendan&longs;i <lb/>dunque i pe&longs;i EF e&longs;&longs;ere in maniera dalla bilancia NO annodati, come &longs;e fo&longs;&longs;e vna <lb/>grandezza &longs;ola d'ambidue ipe&longs;i EF composta, & attacata ne i punti BG, &longs;e dun­<lb/>que &longs;ar anno &longs;cioltii legamenti BG de' pe&longs;i; rimarr anno i pe&longs;i EF <expan abbr="p&etilde;denti">pendenti</expan> da HP; <lb/>&longs;i come prima &longs;tauane in GB. </s> <s>Ma i pe&longs;i EF appiccati in GB pe&longs;ano egualmente <lb/>co'ipe&longs;i LM, & ipe&longs;i EF pendenti dal punto H hanno l'i&longs;te&longs;&longs;a di&longs;po&longs;itione ver<emph.end type="italics"/><lb/> <arrow.to.target n="note86"></arrow.to.target> <emph type="italics"/>&longs;o la bilancia AB, come &longs;e &longs;o&longs;&longs;ero appiccati in BG: Gli isteßi pe&longs;i dunque EF <lb/>pendenti da H pe&longs;aranno egualmente con gli i&longs;te&longs;&longs;i pe&longs;i LM. </s> <s>Sono dunque egual­<lb/>mente graui i pe&longs;i EF attaccati in GB, come attaccati in H.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note83"></margin.target><emph type="italics"/>Per la &longs;ecom da del &longs;esta.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note84"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del quinto<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note85"></margin.target><emph type="italics"/>Per la &longs;esta del primo di Archimede d'lle co&longs;e, che pe&longs;ano egual mente.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note86"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig22"></figure> <figure></figure> <pb pagenum="32"/> <p type="main"> <s><emph type="italics"/>Similmente dimo&longs;treraßi, che i pe&longs;i EF pe&longs;eranno tanto appiccati in qual &longs;i voglia al­<lb/>tro punto, quanto &longs;e l'vno, & l'altro &longs;o&longs;&longs;e pendente dal punto H della diui&longs;ione. </s> <s><lb/>Percioche&longs;e, come di &longs;opra habbiamo in&longs;egnato, &longs;i troueranno ipe&longs;i nella bilancia, à <lb/>i quali i pe&longs;i EF pe&longs;ino egualmente; gliisteßi pe&longs;i EF pendenti da H pe&longs;eranno <lb/>egualmente co' mede&longs;imi pe&longs;i trouati; per e&longs;&longs;ere il punto P &longs;empre il centro della <lb/>grauezzaloro; & la HP a piombo dell'orizonte.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE VI.</s> </p> <p type="main"> <s>I pe&longs;i eguali nella bilancia appiccati hanno in grauezza quella pro­<lb/>portione, che hanno le di&longs;tanze, dalle quali &longs;tanno pendenti. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia la bilancia BAC &longs;o&longs;pe&longs;a nel punto A; & &longs;ia &longs;egata la AC, come pare in D. & <lb/>da i punti DC &longs;iano attaccati EF pe&longs;i eguali. </s> <s>Dico, che il pe&longs;o F ver&longs;o il pe&longs;o E ba <lb/>quella proportione in grauezza, che hala di&longs;tanza CA alla di&longs;tanza AD. </s> <s>Per­<lb/>cioche faccia&longs;i come CA ver&longs;o AD, co&longs;i il pe&longs;o F ver&longs;o vn'altro pe&longs;o, che &longs;ia G. </s> <s><lb/>Dico primai p &longs;i GF pendenti dal punto C tanto pe&longs;are, quanto i pe&longs;i EF penden <lb/>ti da punti DC. </s> <s>Tagli&longs;i DC in due parti eguali in H, & da H &longs;iano fatti pendere <lb/>ambidue i pe&longs;i EF. </s> <s>Pe&longs;eranno EF pre&longs;i in&longs;ieme in quel &longs;ito tanto quanto pe&longs;ano<emph.end type="italics"/> <arrow.to.target n="note87"></arrow.to.target><lb/><emph type="italics"/>in DC. </s> <s>Ponga&longs;i BA eguale ad AH, & &longs;itagli BA in K, di modo, che KA <lb/>&longs;ia eguale ad AD: dapoi dal punto B &longs;ia &longs;atto pendente il pe&longs;o L, ilquale &longs;ia il dop <lb/>pio del pe&longs;o F, cioè eguale a i due pe&longs;i EF, ilqual pe&longs;erà egualmente co'pe&longs;i EF ap <lb/>piccati in H, cioè appiccati in DC. </s> <s>Percioche dunque, come CA ver&longs;o AD, così è <lb/>ilpe&longs;o F ver&longs;o il pe&longs;o G, &longs;arà componendo come CA AD ver&longs;o AD, cioè come <lb/>CK ver&longs;o AD, così ipe&longs;i FG ver&longs;o il pe&longs;o G. </s> <s>Ma per e&longs;&longs;er come CA ver&longs;o AD,<emph.end type="italics"/> <arrow.to.target n="note88"></arrow.to.target><lb/><emph type="italics"/>così il pe&longs;o F al pe&longs;o G, &longs;arà anche conuertendo, come DA ver&longs;o AC, così il pe&longs;o <lb/>G ver&longs;o il pe&longs;o F; & i doppi de icon&longs;eguenti, come DA alla doppia di e&longs;&longs;a AC, <lb/>così il pe&longs;o G al doppio del pe&longs;o F, cioè al pe&longs;o L. </s> <s>Per laqual co&longs;a come CK ver&longs;o<emph.end type="italics"/> <arrow.to.target n="note89"></arrow.to.target><lb/><emph type="italics"/>DA, così ipe&longs;i FG al pe&longs;o G; & come AD alla doppia di AC, così il pe&longs;o G al <lb/>pe&longs;o L, adunque dalla egual proportione come CK alla doppia di AC, così ipe&longs;i FG <lb/>al pe&longs;o L. </s> <s>Ma come CK alla doppia di AC, così la metà di CK, cioè AH, cioè<emph.end type="italics"/> <arrow.to.target n="note90"></arrow.to.target><lb/><emph type="italics"/>BA ver&longs;o AC. </s> <s>Adunque come BA ver&longs;o AC, così FG pe&longs;i al pe&longs;o L. </s> <s>Per laqual<emph.end type="italics"/> <pb/><emph type="italics"/>co&longs;a per la &longs;e&longs;ta dell'i&longs;te&longs;&longs;o primo di Archimede, i due pe&longs;i FG pendenti dal punto C<emph.end type="italics"/><lb/> <arrow.to.target n="note91"></arrow.to.target> <emph type="italics"/>pe&longs;eranno tanto, quanto il pe&longs;o L pendente dal B; cioè quanto i pe&longs;i EF pen­<lb/>denti da i punti DC. </s> <s>Così percioche i pe&longs;i FG tanto pe&longs;ano quanto ipe&longs;i EF, <lb/>leuato via il pe&longs;o comune F, tanto pe&longs;er à il pe&longs;o G appicato in C, quanto il pe<emph.end type="italics"/><lb/> <arrow.to.target n="fig23"></arrow.to.target><lb/><emph type="italics"/>&longs;o E in D. </s> <s>Et perciò il pe&longs;o F al pe&longs;o E hà quella proportione in grauezza, <lb/>che hà al pe&longs;o G. </s> <s>Ma il pe&longs;o F ver&longs;o il G era come CA ver&longs;o AD. adun <lb/>que il pe&longs;o F ancora ver&longs;o il pe&longs;o E hauerà quella proportione in grauczza, che <lb/>ha CA ver&longs;o AD che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note87"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note88"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note89"></margin.target><emph type="italics"/>Per la con&longs;e guenza della quartadel quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note90"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 22. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note91"></margin.target><emph type="italics"/>Per la &longs;ettima del<emph.end type="italics"/> 5.</s> </p> <figure id="fig23"></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e nella bilancia BAC &longs;i faranno pendenti da i punti BC, i pe&longs;i EF eguali; <lb/>Dico &longs;imilmente, che il pe&longs;o E ver&longs;o il pe&longs;o F hà quella proportione in grauezza, <lb/>che ha la di&longs;tanza <lb/>CA alla di&longs;tanza <lb/>AB. </s> <s>faccia&longs;i AD <lb/>eguale ad AB, & <lb/>dal punto D &longs;ia <lb/>fatto <expan abbr="pēdente">pendente</expan> il pe <lb/>&longs;o G eguale al pe <lb/>&longs;o F, ilquale <expan abbr="etiā-">etian-</expan><emph.end type="italics"/><lb/> <arrow.to.target n="fig24"></arrow.to.target><lb/><emph type="italics"/>dio &longs;arà eguale ad E. </s> <s>Et percioche AD è eguale ad AB; i pe&longs;i FG pe&longs;eran <lb/>no egualmente, & hauranno la mede&longs;ima grauezza. </s> <s>Et concio&longs;ia, che la grauezza <lb/>del pe&longs;o E ver&longs;o la grauezza del pe&longs;o G &longs;ia come CA ad AD; &longs;arà la gra­<lb/>uezza del pe&longs;o E ver&longs;o la grauezza del pe&longs;o F, come CA ad AD, cioè CA <lb/>ad AB, che parimente era da mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig24"></figure> <p type="head"> <s>Altramente.</s> </p> <p type="main"> <s><emph type="italics"/>Sia la bilancia BAC, col&longs;uo centro A: & ne i punti BC &longs;iano appiccati pe&longs;i <lb/>eguali GF, & &longs;ia prima il centro A, come &longs;i vuole, fra B, & C. </s> <s>Dico, che <lb/>il pe&longs;o F ver&longs;o il pe&longs;o G hà quella proportione in grauezza, che ha la di&longs;tanza <lb/>CA alla di&longs;tanza AB. </s> <s>Faccia&longs;i come BA ver&longs;o AC, co&longs;i il pe&longs;o F ad vn-<emph.end type="italics"/> <pb pagenum="33"/><emph type="italics"/>altro H, ilquale &longs;ia appiccato in B: i pe&longs;i HF pe&longs;eranno egualmente de A.<emph.end type="italics"/> <arrow.to.target n="note92"></arrow.to.target><lb/><emph type="italics"/>Ma e&longs;&longs;endo i pe&longs;i FG eguali, haurà il pe&longs;o H ver&longs;o il pe&longs;o G la proportione me <lb/>de&longs;ima, che ha ad F. </s> <s>Come dunque CA ver&longs;o AB, co&longs;i è H ver&longs;o G: & <lb/>come H ver&longs;o G, co&longs;i è lagrauezza di H alla grauezza di G, per e&longs;&longs;ere attac <lb/>cati nell i&longs;te&longs;&longs;o punto B. </s> <s>Per laqual co&longs;a come CA ad AB, co&longs;i la grauezza <lb/>del pe&longs;o H alla grauezza del pe&longs;o G. </s> <s>Et concio&longs;ia che la grauezza del pe&longs;o F <lb/>attacato in G &longs;ia<emph.end type="italics"/> <arrow.to.target n="note93"></arrow.to.target><lb/><emph type="italics"/>eguale alla grauez­<lb/>za del pe&longs;o Hattac <lb/>cato in B, &longs;arà la <lb/>grauezza del pe&longs;o F <lb/>ver&longs;o la grauezza <lb/>del pe&longs;o G, come <lb/>CA ver&longs;o AB, <lb/>cioè come la di&longs;tan­<lb/>za alla di&longs;tanza, che <lb/>bi&longs;ognaua mostrare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note92"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del prime di Ar chimede del le co&longs;e che pe &longs;ano egualmente.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note93"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e la bilancia BAC fo&longs;&longs;e tagliata, come &longs;i vuole in D, & appicchin&longs;i in DC <lb/>i pe&longs;i EF eguali. </s> <s>Dico &longs;imilmente co&longs;i e&longs;&longs;ere la grauezza del pe&longs;o F alla gra­<lb/>uezza del pe&longs;o E, come la di&longs;tanza CA alla di&longs;tanza AD. </s> <s>Faccia&longs;i AB <lb/>eguale ad AD <lb/>& &longs;ia appicca­<lb/>to in B il pe&longs;o <lb/>G eguale al pe <lb/>&longs;o E, & alpe <lb/>&longs;o F. </s> <s>Hor <lb/>percioche AB <lb/>è eguale ad A <lb/>D; ipe&longs;i GE<emph.end type="italics"/><lb/> <arrow.to.target n="fig25"></arrow.to.target><lb/><emph type="italics"/>pe&longs;eranno egualmente. </s> <s>Ma per e&longs;&longs;ere la grauezza del pe&longs;o F ver&longs;o la grauezza <lb/>del pe&longs;o G, come CA ad AB, & la grauezza del pe&longs;o E &longs;ia eguale alla <lb/>grauezza del pe&longs;o G; &longs;arà la grauezza del pe&longs;o F ver&longs;o la grauezza del pe&longs;o E, <lb/>come CA ad AB, cioè CA ad AD, che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig25"></figure> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Da que&longs;to è manife&longs;to, che quanto il pe&longs;o è piu di&longs;tante dal centro <lb/>della bilancia, tanto egli è anco piu graue, & per con&longs;eguente mo­<lb/>uer&longs;i piu velocemente. </s> </p> <p type="main"> <s>Quinci oltre à ciò &longs;i mo&longs;trerà facilmente anche la ragione della Sta­<lb/>dera. </s> </p> <pb/> <p type="main"> <s>Corollario vocabolo Latino co&longs;tumato da tutti gli altri Scrittori Italiani in cotal ma <lb/>teria, nè di&longs;piacque à Dante nel 28. cap. </s> <s>del Purgatorio. </s> <s>“Dirotti vn corollario an­<lb/>co per gratia.” vuol dire, &longs;econdo Varrone nel primo libro della lingua Latina, <lb/>quella giunta, & quel &longs;opra piu, che &longs;i dà oltre al pagamento, quando &longs;i comp era <lb/>qualche co&longs;a. </s> <s>Al tempo antico allhor che i recitatori di Tragedie, Comedie, & <lb/>altri Poemi nelle &longs;cene &longs;i portauano bene, & piaceuano à gli vditori, era loro do­<lb/>nato oltra al prezzo a&longs;&longs;egnato, vn corollario per cia&longs;cuno, cioè vna piccola coro <lb/>na per douer&longs;ene ornare le tempie per giunta, & &longs;opra piu delle &longs;ue mercedi. </s> <s>Co&longs;i <lb/>nelle &longs;cienze matematiche v&longs;a&longs;i di aggiungere certe co&longs;e, oltra le propo&longs;itioni, <lb/>qua&longs;i giunte & con&longs;equenze, le quali na&longs;cono dalle co&longs;e primieramente dimo&longs;tra­<lb/>te, & &longs;ono loro corri&longs;pondenti, & non &longs;ono però nè propo&longs;itioni, nè problemi, <lb/>nè lemmi, ma alla &longs;embianza predetta chiaman&longs;i coro llarij, molti de i quali han­<lb/>no congiunta la &longs;ua dimo&longs;tratione. </s> </p> <p type="main"> <s><emph type="italics"/>Hor &longs;ia AB il fusto della Stadera, la cui trutina &longs;ia in C; & &longs;ia il marco della &longs;ta<emph.end type="italics"/><lb/> <arrow.to.target n="note94"></arrow.to.target> <emph type="italics"/>dera E. </s> <s>Appicchi&longs;i in A il pe&longs;o D, che pe&longs;i egualmente col marco E appic­<lb/>cato in F. </s> <s>Appicchi&longs;i parimente vn'altro pe&longs;o G in A, ilqual anco pe&longs;i egual­<lb/>mente col marco E appiccato in B. </s> <s>Dico, la grauezza del pe&longs;o D ver&longs;o la gra­<lb/>uezza del <lb/>G e&longs;&longs;ere co <lb/>&longs;i, come CF <lb/>ver&longs;o CB. </s> <s><lb/>Hor per­<lb/>cioche la <lb/>grauezza <lb/>del pe&longs;o D <lb/>è eguale al <lb/>la grauez­<lb/>za del pe­<lb/>&longs;o E at­<lb/>taccato in <lb/>F, & la<emph.end type="italics"/><lb/> <arrow.to.target n="fig26"></arrow.to.target><lb/><emph type="italics"/>grauezza del pe&longs;o G è eguale alla grauezza del pe&longs;o E po&longs;to in B; &longs;arà la grauez­<lb/>za del pe&longs;o D alla grauezza del pe&longs;o E po&longs;to in F, come la grauezza del pe&longs;o G alla <lb/>grauezza del pe&longs;o E po&longs;to in B; & permutando come la grauezza del pe&longs;o D alla <lb/>grauezza del pe&longs;o G, co&longs;i la grauezza di E po&longs;to in F alla grauezza di E po&longs;to in B; <lb/>ma la grauezza del pe&longs;o E in F alla grauezza di E in B po&longs;to è come CF <lb/>ver&longs;o CB; come dunque la grauezza del pe&longs;o D alla grauezza del pe&longs;o G, co&longs;i <lb/>è CF ver&longs;o CB. </s> <s>Se dunque la parte del fu&longs;to CB diuidera&longs;&longs;i in parti eguali, po <lb/>&longs;to &longs;olo il pe&longs;o E & piu da pre&longs;&longs;o, & piu da lontano dal punto C; le grauezze de <lb/>pe&longs;i, lequali&longs;tanno pendenti dal punto A &longs;aranno traloro manife&longs;te & note. </s> <s>Co­<lb/>me&longs;e la di&longs;tanza CB &longs;arà tripla della di&longs;tanza CF, &longs;arà parimente la grauezza <lb/>di e&longs;&longs;o G tripla della grauezza di D, che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <pb pagenum="34"/> <p type="margin"> <s><margin.target id="note94"></margin.target><emph type="italics"/>Ragione del la stadera.<emph.end type="italics"/></s> </p> <figure id="fig26"></figure> <p type="main"> <s>In altro modo po&longs;&longs;iamo anco v&longs;are la &longs;tadera, affine che le grauezze <lb/>de ipe&longs;i &longs;i facciano note. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il fu&longs;to della stadera AB, la cui trutina &longs;ia in C, & &longs;ia il marco della stadera <lb/>E, ilquale &longs;ia appiccato in A; & &longs;iano i pe&longs;i DG di&longs;uguali, le proportioni delle <lb/>grauezze de quali cerchia­<lb/>mo: &longs;ia appiccato il pe&longs;o D <lb/>in B talche pe&longs;i egual­<lb/>mente con E. </s> <s>Similmente <lb/>appicchi&longs;i il pe&longs;o G in F, <lb/>ilquale pe&longs;i egualmente con <lb/>l'i&longs;te&longs;&longs;o pe&longs;o E. </s> <s>Dico D <lb/>ver&longs;o G co&longs;i e&longs;&longs;ere; come<emph.end type="italics"/> <arrow.to.target n="note95"></arrow.to.target><lb/><emph type="italics"/>CF ver&longs;o CB. </s> <s>Hor perche <lb/>i pe&longs;i DE pe&longs;ano <expan abbr="egualmē">egualmen</expan><emph.end type="italics"/><lb/> <arrow.to.target n="fig27"></arrow.to.target><lb/><emph type="italics"/>te, &longs;arà D ad E, come CA à CB. </s> <s>& concio&longs;ia, che anche i pe&longs;i GE pe&longs;i­<lb/>no egualmente, &longs;arà il pe&longs;o E ver&longs;o il pe&longs;o G, come FC à CA; Per laqual <lb/>co&longs;a per la proportion eguale il pe&longs;o D al pe&longs;o G, co&longs;i &longs;arà, come CF à CB. <lb/>che parimente bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/> <arrow.to.target n="note96"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note95"></margin.target><emph type="italics"/>Per la &longs;esta del primo di Archimede d'lle co&longs;e, che pe&longs;ano egual mente.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note96"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 23. <emph type="italics"/>del quinto<emph.end type="italics"/></s> </p> <figure id="fig27"></figure> <p type="head"> <s>PROPOSITIONE VII.</s> </p> <p type="head"> <s>PROBLEMA.</s> </p> <p type="main"> <s>Dati quanti &longs;i vogliano pe&longs;i nella bilancia, appiccati in qualluogo &longs;i <lb/>&longs;ia, ritrouare il centro della bilancia, dal quale &longs;e &longs;arà fatta penden­<lb/>te la bilancia, i dati pe&longs;i &longs;taranno fermi. </s> </p> <p type="main"> <s>PROBLEMA. </s> <s>Sotto il nome di Propo&longs;itione &longs;i contiene il Problema ancora vo­<lb/>cabolo greco; ma il Problema ha dauantaggio della Propo&longs;itione in particolare, <lb/>che ordina, & in&longs;egna ad operare qualche effetto; doue la Propo&longs;itione &longs;uole &longs;ta <lb/>re nella nuda &longs;peculatione &longs;olamente. </s> <s>Et que&longs;ta è la differenza tra la Propo&longs;itio­<lb/>ne, & il Problema. </s> </p> <pb/> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia la bilancia AB, & &longs;iano dati quanti &longs;i vogliano pe&longs;i CDEFG prendan&longs;i nel <lb/>la bilancia, a piacere i punti AHKLB, da quali &longs;ian fatti pendenti i dati pe&longs;i. </s> <s><lb/>Bi&longs;ogna ritrouar il centro della bilancia, dal quale &longs;e &longs;i far à l'appiccamento, rim anga <lb/>no i dati pe&longs;i. </s> <s>Diuida&longs;i AH in M, &longs;i che HM ad MA &longs;ia come la grauezza <lb/>del pe&longs;o C alla grauezza del pe&longs;o D. </s> <s>Dapoi diuida&longs;i anco BL in N, &longs;i che <lb/>LN ad NB &longs;ia come la grauez <gap/> pe&longs;o G alla grauezza del pe&longs;o F. </s> <s>Et di­<lb/>uida&longs;i MN in O, &longs;i che MO ver&longs;o ON &longs;ia come la grauezza de pe&longs;i FG <lb/>alla grauezza de'pe&longs;i CD. </s> <s>Et in fine diuida&longs;i KO in P, &longs;i che KP ver&longs;o PO <lb/>&longs;ia come la grauezza de'pe&longs;i CD FG alla grauezza del pe&longs;o E. </s> <s>Hor percio-<emph.end type="italics"/><lb/> <arrow.to.target n="note97"></arrow.to.target> <emph type="italics"/>che i pe&longs;i CDFG tanto pe&longs;ano in O, quanto CD in M, & FG in N; <lb/>pe&longs;eranno egualmente i pe&longs;i CD in M, & FG in N, & il pe&longs;o E in K, <lb/>&longs;e &longs;aranno &longs;o&longs;pe&longs;i nel punto P. </s> <s>Et concio&longs;ia, che ipe&longs;i CD tanto pe&longs;ino in M, <lb/>quanto in AH, & FG in N quanto in LB; ipe&longs;i CDFG pendenti da' <lb/>punti AHLB, & il pe&longs;o E da K, &longs;e da P &longs;aranno &longs;o&longs;pe&longs;i, pe&longs;eranno egual­<lb/>mente, & rimarranno. </s> <s>egli è dunque trouato il P centro della bilancia, dalquale <lb/>rimangono i pe&longs;i dati. </s> <s>Che bi&longs;ogna operare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note97"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Da que&longs;to è chiaro, che &longs;ei centri della grauezza de' pe&longs;i CDEFG <lb/>fo&longs;&longs;ero ne' punti AHKLB, &longs;arebbe il punto P il centro della <lb/>grauezza della magnitudine compo&longs;ta di tutti i pe&longs;i CDEFG. </s> </p> <p type="main"> <s><emph type="italics"/>Que&longs;to è manife&longs;to dalla diffinitione del centro della grauezza, concio&longs;ia che i pe&longs;i ri­<lb/>mangano, &longs;e &longs;ono &longs;o&longs;tenuti dal punto P.<emph.end type="italics"/></s> </p> <p type="head"> <s><emph type="italics"/>Il fine della Bilancia.<emph.end type="italics"/></s> </p> <pb pagenum="35"/> <p type="head"> <s>DELLA LEVA.</s> </p> <figure></figure> <p type="head"> <s>LEMMA.</s> </p> <p type="main"> <s>Siano quattro grandezze ABCD; & &longs;ia la A <lb/>maggiore della B, & C maggiore della D. </s> <s>Dico, <lb/>che A ver&longs;o D hà proportione maggiore di quello <lb/>che hà B ver&longs;o C. </s> </p> <p type="main"> <s><emph type="italics"/>Hor percioche A ver&longs;o C hà proportion maggio­<lb/>re, che B ver&longs;o C; & A parimente ver&longs;o D<emph.end type="italics"/> <arrow.to.target n="note98"></arrow.to.target><lb/><emph type="italics"/>hà proportion maggiore di quel che ha ver&longs;o C: <lb/>Dunque A ver&longs;o D l'hauer à maggiore, che B <lb/>ver&longs;o C, Che bi&longs;ognaua mostrare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note98"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>PROPOSITIONE I.</s> </p> <p type="main"> <s>La po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o attaccato alla Leua, ha la proportio <lb/>ne mede&longs;ima al detto pe&longs;o, che ha la di&longs;tanza della Leua fra il &longs;o&longs;te <lb/>gno po&longs;ta, & lo attaccamento del pe&longs;o, alla di&longs;tanza, che è dal &longs;o&longs;te <lb/>gno alla po&longs;&longs;anza. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB, il cui &longs;oftegno &longs;ia C; & &longs;iail pe&longs;o D pendente da A con AH, <lb/>&longs;i che AH &longs;ia &longs;empre à piombo dell'orizonte: & &longs;ia la po&longs;&longs;anza &longs;oftenente il pe-<emph.end type="italics"/><lb/> <arrow.to.target n="fig28"></arrow.to.target><lb/><emph type="italics"/>&longs;o in B. </s> <s>Dico che la po&longs;&longs;anza posta in B ver&longs;o il pe&longs;o D &longs;ta co&longs;i, come la CA<emph.end type="italics"/> <pb/> <arrow.to.target n="note99"></arrow.to.target> <emph type="italics"/>ver&longs;o la CB. </s> <s>Faccia&longs;i come la BC alla CA, co&longs;i il pe&longs;o D ad vn'altro pe&longs;o <lb/>E, talche &longs;e egli in B &longs;arà appiccato, pe&longs;erà <expan abbr="egualm&etilde;te">egualmente</expan> con D, per e&longs;&longs;er il C cen <lb/>tro della grauezza di ambidue. </s> <s>Per laqual co&longs;a vna po&longs;&longs;anza eguale ad e&longs;&longs;o E po <lb/>&longs;ta nel <lb/>mede&longs;i <lb/>mo lo <lb/>go pe­<lb/>&longs;erà e­<lb/>gual­<lb/>mente <lb/>con e&longs;-<emph.end type="italics"/><lb/> <arrow.to.target n="fig29"></arrow.to.target><lb/><emph type="italics"/>&longs;o D, nella leua AB, collocando il &longs;o&longs;tegno &longs;uo in C, cioè impedirà, che il pe-<emph.end type="italics"/><lb/> <arrow.to.target n="note100"></arrow.to.target> <emph type="italics"/>&longs;o D non inchini in giu&longs;o, &longs;i come impedi&longs;ce il pe&longs;o E. </s> <s>Ma la po&longs;&longs;anza di B al <lb/>pe&longs;o D hàla mede&longs;ima proportione, che il pe&longs;o E ha all'iste&longs;&longs;o D: adunque la <lb/>po&longs;&longs;anza di B ver&longs;o il pe&longs;o D &longs;arà come CA ver&longs;o CB; cioè la di&longs;tanza del­<lb/>la leua dal &longs;ostegno al &longs;o&longs;tenimento del pe&longs;o, alla di&longs;tanza dal &longs;ostegno alla po&longs;&longs;an­<lb/>za, che bi&longs;o gnaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note99"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del<emph.end type="italics"/> 1. <emph type="italics"/>di Archi mede delleco &longs;ache egual <expan abbr="m&etilde;te">mente</expan> pe&longs;ano.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note100"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure id="fig28"></figure> <figure id="fig29"></figure> <p type="main"> <s>Di quì ageuolmente &longs;i puote mo&longs;trare, che <expan abbr="quãto">quanto</expan> il &longs;o&longs;tegno &longs;arà piu <lb/>vicino al pe&longs;o, tanto minor po&longs;&longs;anza &longs;i ricerca à &longs;o&longs;tenere il detto <lb/>pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Poste le co&longs;e mede&longs;ime &longs;ia il &longs;o&longs;tegno in F piu da pre&longs;&longs;o ad A, che C; & faccia&longs;i <lb/>come BF ad FA, co&longs;i il pe&longs;o D ad vn'altro pe&longs;o G, ilquale &longs;e in B &longs;ia ap-<emph.end type="italics"/><lb/> <arrow.to.target n="note101"></arrow.to.target> <emph type="italics"/>piccato; i pe&longs;i DG dal &longs;o&longs;tegno F pe&longs;eranno egualmente. </s> <s>Hor percioche BF <lb/>è mag­<lb/>giore di <lb/>BC, & <lb/>CA<emph.end type="italics"/><lb/> <arrow.to.target n="note102"></arrow.to.target> <emph type="italics"/>maggio <lb/>re di AF; <lb/>la <lb/>propor­<lb/>tione di<emph.end type="italics"/><lb/> <arrow.to.target n="fig30"></arrow.to.target><lb/><emph type="italics"/>BF ver&longs;o FA &longs;arà maggiore, che di BC ver&longs;o CA: & perciò maggiore anco <lb/>&longs;arà la proportione del pe&longs;o D alpe&longs;o G, che de l'iste&longs;&longs;o D ad E: Dunque il<emph.end type="italics"/><lb/> <arrow.to.target n="note103"></arrow.to.target> <emph type="italics"/>pe&longs;o G &longs;arà minore del pe&longs;o E. </s> <s>& concio&longs;ia che la po&longs;&longs;anza po&longs;ta in B eguale à <lb/>G pe&longs;i egualmente con D, auerrà, che minore po&longs;&longs;anza di quella, laquale è eguale <lb/>al pe&longs;o E &longs;o&longs;tenter à il pe&longs;o D; e&longs;&longs;endo la leua AB, & il &longs;o&longs;tegno &longs;uo doue è F, <lb/>che &longs;e egli &longs;o&longs;&longs;e doue è C. </s> <s>Similmente anche mo&longs;trera&longs;&longs;i, che quanto piu dapre&longs;&longs;o &longs;a <lb/>rà il &longs;o&longs;tegno al pe&longs;o D, &longs;empre vi &longs;i ricercherà anco po&longs;&longs;anza minore per &longs;o&longs;tentare <lb/>il detto pe&longs;o D.<emph.end type="italics"/></s> </p> <pb pagenum="36"/> <p type="margin"> <s><margin.target id="note101"></margin.target><emph type="italics"/>Per la mede &longs;ima &longs;esta.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note102"></margin.target><emph type="italics"/>Per lo Lemma.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note103"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto<emph.end type="italics"/></s> </p> <figure id="fig30"></figure> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Onde &longs;i puote raccogliere chiaramente, che e&longs;&longs;endo AF minore di <lb/>FB, minor po&longs;&longs;anza anco &longs;i ricerca in B per &longs;o&longs;tenere il pe&longs;o D. <lb/>& e&longs;&longs;endo eguale, eguale: & maggiore, maggiore. </s> </p> <p type="head"> <s>PROPOSITIONE II.</s> </p> <p type="main"> <s>In altra maniera po&longs;&longs;iamo v&longs;are la Leua. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia B, & il pe&longs;o C &longs;ia attaccato, come &longs;i vuole, in D<emph.end type="italics"/> <arrow.to.target n="note104"></arrow.to.target><lb/><emph type="italics"/>fra AB; & &longs;ia la po&longs;&longs;anza in A che &longs;ostiene il pe&longs;o C. </s> <s>Dico, che &longs;i come <lb/>BD à BA; co&longs;i è lapo&longs;&longs;anza di A' al pe&longs;o C. </s> <s>Appicchi&longs;i in A il pe&longs;o E <lb/>eguale al C; & come AB ver&longs;o BD, co&longs;i faccia&longs;i il pe&longs;o E ver&longs;o vn'altro pe&longs;o,<emph.end type="italics"/> <arrow.to.target n="note105"></arrow.to.target><lb/><emph type="italics"/>come F. </s> <s>Et percioche i pe&longs;i CE &longs;ono tra&longs;e eguali, &longs;arà il pe&longs;o C ver&longs;o il pe&longs;o F <lb/>come AB ver&longs;o BD. </s> <s>Attacchi&longs;i parimente il pe&longs;o F in A. & percioche il<emph.end type="italics"/> <arrow.to.target n="note106"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o E al pe&longs;o F <lb/>è come la grauez <lb/>za del pe&longs;o di E <lb/>alla grauezza di <lb/>F; & il pe&longs;o E <lb/>ad F è come AB <lb/>à BD; come <expan abbr="dũ">dum</expan> <lb/>que la grauezza <lb/>del pe&longs;o E alla<emph.end type="italics"/> <arrow.to.target n="note107"></arrow.to.target><lb/> <arrow.to.target n="fig31"></arrow.to.target><lb/><emph type="italics"/>grauezza del pe&longs;o F, co&longs;i è AB ver&longs;o BD. </s> <s>ma come AB à BD, co&longs;i è la <lb/>grauezza del pe&longs;o E alla grauezza del pe&longs;o C: Per laqual co&longs;a la grauezza del <lb/>pe&longs;o E alla grauezza del pe&longs;o F co&longs;i &longs;arà, come la grauezza del pe&longs;o E alla gra­<lb/>uezza del pe&longs;o C. </s> <s>I pe&longs;i dunque CF hanno la mede&longs;ima grauezza: &longs;i che pon­<lb/>ga&longs;i la po&longs;&longs;anza di A che &longs;o&longs;tenga il pe&longs;o F, &longs;arà la po&longs;&longs;anza di A eguale al pe&longs;o <lb/>F. & percioche il pe&longs;o E attaccat<gap/> in A è graue egualmente, come il C appicca-<emph.end type="italics"/> <arrow.to.target n="note108"></arrow.to.target><lb/><emph type="italics"/>to in D; hauer à la proportione i&longs;te&longs;&longs;a la po&longs;&longs;anza di A ver&longs;o la grauezza del pe&longs;o <lb/>F appiccato in A, che ha alla grauezza del pe&longs;o C appiccato in D. </s> <s>Mala po&longs;&longs;an <lb/>za di A eguale ad F &longs;o&longs;tiene il pe&longs;o F; dunque la po&longs;&longs;anza di A &longs;o&longs;tenterà anco <lb/>il pe&longs;o C. </s> <s>Et co&longs;i per e&longs;&longs;ere la po&longs;&longs;anza di A eguale al pe&longs;o F, & il pe&longs;o C ver&longs;o <lb/>il pe&longs;o F &longs;ia come AB à BD; &longs;arà il pe&longs;o C ver&longs;o la po&longs;&longs;anza po&longs;ta in A come <lb/>AB à BD. & conuertendo, come BD à BA, co&longs;i la po&longs;&longs;anza po&longs;ta in A ver <lb/>&longs;o il pe&longs;o C. </s> <s>Dunque la po&longs;&longs;anza ver&longs;o il pe&longs;o co&longs;i &longs;arà, come la di&longs;tanza, che è fra<emph.end type="italics"/> <arrow.to.target n="note109"></arrow.to.target><lb/><emph type="italics"/>il &longs;o&longs;tegno, & l'appiccamento del pe&longs;o alla di&longs;tanza, che è dal &longs;o&longs;tegno alla po&longs;&longs;an­<lb/>za, che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <pb/> <p type="margin"> <s><margin.target id="note104"></margin.target><emph type="italics"/>Nella &longs;esta di questo de la bilancia.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note105"></margin.target><emph type="italics"/>Dalla<emph.end type="italics"/> 11. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note106"></margin.target><emph type="italics"/>Per la &longs;esta della <expan abbr="bilācia">bilancia</expan><emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note107"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note108"></margin.target><emph type="italics"/>Per la &longs;ettima del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note109"></margin.target><emph type="italics"/>Per lo Corol lario della<emph.end type="italics"/> 4 <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure id="fig31"></figure> <p type="head"> <s>Altramente.</s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia B, & il pe&longs;o E &longs;ia pendente dal punto C, & <lb/>&longs;ia in A la forza, che &longs;ostiene l pe&longs;o E. </s> <s>Dico, che &longs;i come BC à BA, co&longs;i è<emph.end type="italics"/><lb/> <arrow.to.target n="fig32"></arrow.to.target><lb/><emph type="italics"/>anco la po&longs;&longs;anza di A ver&longs;o il pe&longs;o E. </s> <s>Allunghi&longs;i AB in D, & faccia&longs;i <lb/>BD eguale à BC; & appicchi&longs;i il pe&longs;o F al punto D, che &longs;ia eguale al pe&longs;o E; <lb/>& parimente dal punto A &longs;i faccia pendere il punto G in modo, che il pe&longs;o F hab <lb/>bia la proportione i&longs;te&longs;&longs;a ver&longs;o il pe&longs;o G, che ha AB à BD. </s> <s>ipe&longs;i FG verranno <lb/>à pe&longs;ar egualmente: & concio&longs;ia che CB &longs;ia eguale à BD, anco i pe&longs;i FE egua <lb/>li pe&longs;eranno egualmente. </s> <s>Ma ipe&longs;i FEG nella bilancia, ouero nella leua DBA <lb/>appiccati, il cui &longs;o&longs;tegno è B, non pe&longs;eranno egualmente, ma inchineranno à ba&longs;&longs;o <lb/>dalla parte di A. </s> <s>Per laqual co&longs;a ponga&longs;i in A tanta forza, che ipe&longs;i FEG pe&longs;i­<lb/>no egualmente, &longs;arà la po&longs;&longs;anza in A eguale al pe&longs;o G; peroche i pe&longs;i FE pe&longs;a­<lb/>no egualmente, & la forza in A niente altro deue fare, che &longs;o&longs;tenere il pe&longs;o G, ac­<lb/>cio che non de&longs;cenda. </s> <s>Et percio che i pe&longs;i FEG, & la po&longs;&longs;anza in A pe&longs;ano egual <lb/>mente, leuati dunque via i pe&longs;i FG, i quali pe&longs;ano egualmente, i re&longs;tanti pe&longs;eran­<lb/>no pur egualmente, cioè la po&longs;&longs;anza in A co'l pe&longs;o E, cioè la po&longs;&longs;anza in A &longs;o­<lb/>sterra ilpe&longs;o E, &longs;i che la leua AB rimanga, come era prima. </s> <s>Et per e&longs;&longs;ere la <lb/>po&longs;&longs;anza in A eguale al pe&longs;o G, & il pe&longs;o E eguale al pe&longs;o F, haurà la po&longs;&longs;anza <lb/>in A la proportione iste&longs;&longs;a al pe&longs;o E, che hà BD, cioè BC à BA, che bi&longs;ogna <lb/>ua mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig32"></figure> <p type="head"> <s>COROLLARIO I.</s> </p> <p type="main"> <s>Da que&longs;to etiandio, come prima, puote e&longs;&longs;ere manife&longs;to, che &longs;eil pe&longs;o <lb/>E &longs;arà po&longs;to piu vicino al &longs;o&longs;tegno B, come in H, minore <lb/>po&longs;&longs;anza po&longs;ta in A puote &longs;o&longs;tener il detto pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche minor proportione ha HB à BA, che CB à BA. & quanto piu da<emph.end type="italics"/><lb/> <arrow.to.target n="note110"></arrow.to.target> <emph type="italics"/>vicino il pe&longs;o &longs;arà al &longs;o&longs;tegno, &longs;empre anco &longs;i mo&longs;trerà &longs;imilmente minor po&longs;&longs;anza <lb/>poter &longs;o&longs;tener il pe&longs;o E.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note110"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO II.</s> </p> <p type="main"> <s>Segue etiandio, che la po&longs;&longs;anza in A &longs;empre è minore del pe&longs;o E: </s> </p> <pb pagenum="37"/> <p type="main"> <s><emph type="italics"/>Percioche pigli&longs;i tra A & B qual punto &longs;i voglia, come C, &longs;empre BC &longs;arà <lb/>minore di BA.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO III.</s> </p> <p type="main"> <s>Da que&longs;to parimente &longs;i puote cauare, che &longs;e due &longs;aranno le po&longs;&longs;anze, <lb/>l'vna in A, & l'altra in B, & ambedue &longs;o&longs;tentino il pe&longs;o E, la po&longs;­<lb/>&longs;anza in A ver&longs;o la po&longs;&longs;anza in B è come BC ver&longs;o CA. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche la leua BA fal'officio di due leue, & AB &longs;ono come due &longs;o&longs;tegni, cioè <lb/>quando AB è leua, & la forza che &longs;o&longs;tiene è in A, &longs;arà il &longs;uo &longs;o&longs;tegno B. </s> <s>Ma <lb/>quando BA è leua, & la po&longs;&longs;anza &longs;ta in B, il &longs;o&longs;tegno &longs;arà A, & il pe&longs;o <lb/>&longs;empre rimane appicca­<lb/>to in C. </s> <s>Et percio che la <lb/>po&longs;&longs;anza in A ver&longs;o il <lb/>pe&longs;o E è come BC à<emph.end type="italics"/> <arrow.to.target n="note111"></arrow.to.target><lb/><emph type="italics"/>BA, & come il pe&longs;o <lb/>E alla po&longs;&longs;anza, che è <lb/>in B, co&longs;i è BA ad <lb/>AC, &longs;arà per la propor<emph.end type="italics"/><lb/> <arrow.to.target n="fig33"></arrow.to.target><lb/><emph type="italics"/>tion eguale la po&longs;&longs;anza in A alla po&longs;&longs;anza in B come BC à CA, & à que <lb/>&longs;to modo facilmente ancora potremo cono&longs;cere la proportione, laquale è po&longs;ta de <lb/>Ari&longs;totele nelle que&longs;tioni Mecaniche alla que&longs;tione 29.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note111"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 22. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure id="fig33"></figure> <p type="head"> <s>COROLLARIO IIII.</s> </p> <p type="main"> <s>E manife&longs;to etiandio, che ambedue le po&longs;&longs;anze in A, & in B <lb/>pre&longs;e in&longs;ieme, &longs;ono eguali al pe&longs;o E. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche il pe&longs;o E alla po&longs;&longs;anza in A è come BA à BC, & l'i&longs;te&longs;&longs;o pe&longs;o E <lb/>ver&longs;o la po&longs;&longs;anza in B è come BA ad AC; Per laqual co&longs;a il pe&longs;o E ver­<lb/>&longs;o l'vna, & l'altra po&longs;&longs;anzain A, & in B pre&longs;e in&longs;ieme, è come AB ver&longs;o <lb/>BC, & CA in&longs;ieme, cioè ver&longs;o BA. </s> <s>il pe&longs;o dunque E è eguale ad amen­<lb/>due le po&longs;&longs;anze pre&longs;e in&longs;ieme.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE III.</s> </p> <p type="main"> <s>In altro modo ancora po&longs;&longs;iamo v&longs;are la Leua. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia B. & &longs;ia il pe&longs;o C appiccato al punto A, <lb/>& &longs;ia la po&longs;&longs;anza in D, comunque &longs;i voglia tra AB, &longs;o&longs;tenente il pe&longs;o C. </s> <s>Di­<lb/>co che come AB à BD, co&longs;i è la po&longs;&longs;anza in D al pe&longs;o C. </s> <s>Appicchi&longs;i al <lb/>punto D il pe&longs;o E eguale à C; & come BD à BA, co&longs;i &longs;accia&longs;i il pe&longs;o <lb/>E ad vn'altro pe&longs;o, come F: & per e&longs;&longs;ere i pe&longs;i CE traloro eguali, &longs;arà an­<lb/>co il pe&longs;o C al <lb/>pe&longs;o F, come <lb/>BD à BA. </s> <s><lb/>Appicchi&longs;i &longs;imil <lb/>mente il pe&longs;o F <lb/>in D. & per­<lb/>che il pe&longs;o E ad <lb/>F è come la gra <lb/>uezza del pe&longs;o <lb/>E alla grauez­<lb/>za del pe&longs;o F; <lb/>& il pe&longs;o E al<emph.end type="italics"/><lb/> <arrow.to.target n="fig34"></arrow.to.target><lb/> <arrow.to.target n="note112"></arrow.to.target> <emph type="italics"/>pe&longs;o F è come BD à BA. </s> <s>Come dunque la grauezza del pe&longs;o E alla gra­<lb/>uezza del pe&longs;o F, co&longs;i è BD à BA. </s> <s>Ma come BD à BA, co&longs;i è la gra­<lb/>uezza del pe&longs;o E alla grauezza del pe&longs;o C. </s> <s>Per laqual co&longs;a la grauezza del<emph.end type="italics"/><lb/> <arrow.to.target n="note113"></arrow.to.target> <emph type="italics"/>pe&longs;o E alla grauezza del pe&longs;o F ha la proportione mede&longs;ima, che ha alla gra­<lb/>uezza del pe&longs;o C. </s> <s>i pe&longs;i dunque CF hanno la grauezza mede&longs;ma. </s> <s>Sia dunque<emph.end type="italics"/><lb/> <arrow.to.target n="note114"></arrow.to.target> <emph type="italics"/>la po&longs;&longs;anza in D &longs;o&longs;tenente il pe&longs;o F, che verrà ad e&longs;&longs;ere la detta po&longs;&longs;anza in <lb/>D eguals al pe&longs;o F. & percioche il pe&longs;o F posto in D è graue egualmente <lb/>come il pe&longs;o C po&longs;to in A; haur à la po&longs;&longs;anza in D la proportione mede&longs;ima <lb/>ver&longs;o la grauezza del pe&longs;o F, che ha alla grauezza del pe&longs;o C. </s> <s>Ma la po&longs;&longs;anza <lb/>in D &longs;o&longs;tiene il pe&longs;o F, dunque la po&longs;&longs;anza in D &longs;o&longs;tenterà anco il pe&longs;o C; & <lb/>il pe&longs;o C alla po&longs;&longs;anza in D &longs;arà co&longs;i come il pe&longs;o C al pe&longs;o F; & C ad F<emph.end type="italics"/><lb/> <arrow.to.target n="note115"></arrow.to.target> <emph type="italics"/>è come BD à BA, &longs;arà dunque il pe&longs;o C alla po&longs;&longs;anza in D, come BD à <lb/>BA: & conuertendo come AB à BD, co&longs;i la po&longs;&longs;anza in D al pe&longs;o C. </s> <s>La <lb/>po&longs;&longs;anza dunque al pe&longs;o, è come la di&longs;tanza dal &longs;ostegno allo appiccamento del pe­<lb/>&longs;o alla distanza dal &longs;o&longs;tegno alla po&longs;&longs;anza. </s> <s>che bi&longs;ognaua mostrare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note112"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo della bilancia.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note113"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo della bilancia.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note114"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note115"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure id="fig34"></figure> <p type="head"> <s>Altramente.</s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia B. & dal punto A &longs;ia fatto pendente il pe&longs;o <lb/>C, & &longs;ia la po&longs;&longs;anza in D &longs;o&longs;tenente il pe&longs;o C. </s> <s>Dico, che come AB à BD, <lb/>co&longs;i è la po&longs;&longs;anza in D al pe&longs;o C. </s> <s>allunghi&longs;ila AB in E, & faccia&longs;i BE egua­<lb/>le à BA, & al punto E &longs;ia appiccato il pe&longs;o F eguale al pe&longs;o C; & come BD à <lb/>BE co&longs;i faccia&longs;i il pe&longs;o F ad vn'altro pe&longs;o G, ilquale &longs;ia appiccato al punto D, <lb/>i pe&longs;i FG pe&longs;eranno egualmente. </s> <s>& percioche AB è eguale à BE, & i pe&longs;i<emph.end type="italics"/> <pb pagenum="19"/><emph type="italics"/>mente non è della grauez <lb/>za cagione? </s> <s>Di questo ef <lb/>fetto mostrano di produ­<lb/>cere in mezo que&longs;ta cagio <lb/>ne, perche CG è la mé­<lb/>ta, & CF la trutina; <lb/>&longs;e (dicono e&longs;&longs;i) CG fo&longs; <lb/>&longs;e la trutina, & CF la <lb/>méta, all'hora l'angolo <lb/>FCE &longs;arebbe cagione <lb/>della grauezza, ma non <lb/>già il DCG ad e&longs;&longs;o e­<lb/>guale laquale ragione è al <lb/>tutto fatta con la imagi­<lb/>natione, & di voglia pro <lb/>pria. </s> <s>Peroche, che puote <lb/>importare che la trutina <lb/>&longs;ia ouero in CF, ouero <lb/>in CG, e&longs;&longs;endo la bilan <lb/>cia DE &longs;empre &longs;o&longs;ten­<lb/>tata nell'i&longs;te&longs;&longs;o punto C? </s> <s>Ma affine che l'inganno loro re&longs;ti più chiaro.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia la mede&longs;ima bilancia AB, il cui mezo C. </s> <s>dapoitutta la FG &longs;ia la trutina, <lb/>laquale &longs;tia immobile, & &longs;o&longs;tenga la bilancia AB nel punto C. </s> <s>& moua&longs;i la <lb/>bilancia in DE. & per­<lb/>cioche la trutina è &longs;opra, & <lb/>&longs;otto la bilancia, quale ango <lb/>lo &longs;arà cagione della grauez <lb/>za, e&longs;&longs;endo &longs;o&longs;tenuta la bi­<lb/>lancia DE &longs;empre nel pun <lb/>to mede&longs;imo? </s> <s>Diranno for­<lb/>&longs;e &longs;e la trutina &longs;arà &longs;o&longs;tenu­<lb/>ta dalla po&longs;&longs;anza po&longs;ta in <lb/>F, allhora CG &longs;arà tan­<lb/>to quanto la méta, & l'an­<lb/>golo DCG &longs;arà della gra <lb/>uezza cagione. </s> <s>Ma&longs;e<emph.end type="italics"/><lb/> <arrow.to.target n="fig35"></arrow.to.target><lb/><emph type="italics"/>egli &longs;arà &longs;ostenuto in G, allhora FCE &longs;arà cagione della grauezza, & la CF <lb/>&longs;arà tanto quanto la méta. </s> <s>della qual co&longs;a niuna cagione pare poter&longs;i addurre, <lb/>&longs;e <expan abbr="nõ">non</expan> imaginata; peroche la méta (che dicono) non pare hauere à modo veruno nien <lb/>te di virtù che tiri dalla parte dell'angolo maggiore alcuna volta, & alcuna dalla <lb/>parte del minore. </s> <s>Ma &longs;ia &longs;o&longs;tenuta la trutina da due po&longs;&longs;anze in F cioè, & in G, <pb/>&longs;arà la inclinatione dell'vno al moui­<lb/>mento in giù, tale parimente &longs;arà la re <lb/>&longs;istenza dell'altro al mouimento in sù. </s> <s><lb/>Adunque il pe&longs;o po&longs;to in E non mo­<lb/>uerà in sù il pe&longs;o po&longs;to in D: ne il pe&longs;o <lb/>po&longs;to in D: &longs;i mouerà in giù &longs;i fatta­<lb/>mente, che moua in sù il pe&longs;o po&longs;to in <lb/>E. </s> <s>imperoche e&longs;&longs;endo l'angolo CEB <lb/>egualea CDA, & l'angolo CEM <lb/>&longs;ia eguale all'angolo CDH; &longs;arà il <lb/>re&longs;tante MEB eguale al re&longs;tante<emph.end type="italics"/><lb/> <arrow.to.target n="note116"></arrow.to.target> <emph type="italics"/>HDA. </s> <s>La &longs;ce&longs;a dunque del pe&longs;o po­<lb/>&longs;to in D &longs;arà eguale alla &longs;alita del pe­<lb/>&longs;o po&longs;to in E. </s> <s>Adunque il pe&longs;o po&longs;to <lb/>in D non mouerà in sù il pe&longs;o po&longs;to <lb/>in E. </s> <s>Dalle quali co&longs;e &longs;egue che i pe&longs;i <lb/>po&longs;ti in DE, in quanto tra loro &longs;o­<lb/>no congiunti, &longs;ono egualmente graui.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note116"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del prime.<emph.end type="italics"/></s> </p> <figure id="fig35"></figure> <figure></figure> <p type="main"> <s><emph type="italics"/>L'altra ragione po&longs;cia, con laquale vorrebbono mo&longs;trare, che &longs;imilmente la bilancia <lb/>DE ritorna in AB, con dire, che e&longs;&longs;endo la trutina della bilancia CF, la méta <lb/>viene ad e&longs;&longs;er CG. & percioche l'angolo DCG è maggiore dell'angolo ECG, <lb/><gap/> pe&longs;o po&longs;to in D &longs;arà più graue del po&longs;to in E; dunque la bilancia DE ritorne <lb/>ra in AB; non conchiude nulla al parer mio; & que&longs;ta fintione della trutina, & <lb/>della méta è più to&longs;to da trala&longs;ciare, & pa&longs;&longs;arla con &longs;ilentio, che farne pur vna paro <lb/>la per confonderla, e&longs;&longs;endo del tutto co&longs;a volontaria, percioche la nece&longs;&longs;aria ragione <lb/>per laquale il pe&longs;o po&longs;to in D dall' angolo maggiore &longs;ia più graue, & perche il mag <lb/>giore angolo &longs;ia cagione di grauezzamaggiore non appare in niun loco. </s> <s>che &longs;e gli <lb/>angoli &longs;aranno tra loro paragonati, e&longs;&longs;endo l'angolo GCD eguale all'angolo <lb/>FCE; &longs;e l'angolo GCD è cau&longs;a della grauezza, perche l'angolo FCE &longs;imil- <pb/>ilche &longs;i puote fare per nece&longs;&longs;ità, come &longs;e la po&longs;&longs;anza posta in F &longs;o&longs;&longs;e tanto debile, <lb/>che per &longs;e &longs;te&longs;&longs;a pote&longs;&longs;e &longs;o&longs;tentare &longs;olamente la metà del pe&longs;o & &longs;ia la po&longs;&longs;anza <lb/>posta in G eguale alla po&longs;&longs;anza po&longs;ta in F, & ambedue in&longs;ieme co' pe&longs;i &longs;o&longs;tenga­<lb/>no la bilancia. </s> <s>all'hora quale angolo &longs;arà cagione della grauezza? </s> <s>non gia <lb/>FCE, peroche la trutina è <lb/>in CF, & è &longs;o&longs;tentata in <lb/>F: ne meno il DCG, e&longs;&longs;en <lb/>do la trutina in CG, & pa <lb/>rimente &longs;o&longs;tentata in G. </s> <s><lb/>Non &longs;aranno dunque gli an <lb/>goli della grauezza cagione. </s> <s><lb/>Co&longs;i ne anche la bilancia <lb/>DE da que&longs;to &longs;ito per que <lb/>&longs;ta cagione &longs;i mouerà. </s> <s>Ma<emph.end type="italics"/><lb/> <arrow.to.target n="note117"></arrow.to.target> <emph type="italics"/>que&longs;ta loro &longs;entenza pare <lb/>e&longs;&longs;ere confermata da e&longs;&longs;i in <lb/>due modi. </s> <s>Primieramente<emph.end type="italics"/><lb/> <arrow.to.target n="fig36"></arrow.to.target><lb/><emph type="italics"/>dicono Ari&longs;totele nelle que&longs;tioni mecaniche hauere propo&longs;to que&longs;te due que&longs;tioni &longs;o <lb/>lamente, & le &longs;ue dimo&longs;trationi e&longs;&longs;ere fondate &longs;i nel maggiore, & nel minore <lb/>angolo, & &longs;i nella giacitura della trutina della bilancia. </s> <s>Affermano dapoi que&longs;to <lb/>iste&longs;&longs;o in&longs;egnare la e&longs;perientia ancora, cioè, che la bilancia DE, &longs;tando la &longs;ua <lb/>trutina in CF, ritorna in AB egualmente di&longs;tante dall'orizonte. </s> <s>& quando <lb/>la trutina &longs;tà in CG, mouer&longs;i in FG. </s> <s>Mane Ari&longs;totele, ne la e&longs;perienza fauo­<lb/>ri&longs;cono que&longs;ta loro opinione, anzi più to&longs;to le &longs;ono contrarij. </s> <s>Peroche in quan­<lb/>to appartiene alla e&longs;perienza &longs;i ingannano, e&longs;&longs;endo mani&longs;e&longs;to ciò per e&longs;perienza <lb/>accadere, all'hor che il centro ancora della bilancia &longs;arà collocato ò &longs;opra, ò &longs;ot­<lb/>to della bilancia, ma non già auenire que&longs;to stando la trutina ò &longs;opra &longs;olamente, <lb/>è &longs;otto.<emph.end type="italics"/></s> </p> <pb pagenum="20"/> <p type="margin"> <s><margin.target id="note117"></margin.target><emph type="italics"/>il Cardano.<emph.end type="italics"/></s> </p> <figure id="fig36"></figure> <p type="main"> <s><emph type="italics"/>Imperoche &longs;e la bilancia A <lb/>B haue&longs;&longs;e il centro C <lb/>&longs;opra la bilancia, & fo&longs;­<lb/>&longs;e la trutina CD &longs;otto <lb/>la bilancia, & &longs;i moue&longs;­<lb/>&longs;e la bilancia in EF, al <lb/>lhora EF di nouo ri­<lb/>tornerà in AB. egual­<lb/>mente di&longs;tante dall'o­<lb/>rizonte. </s> <s>&longs;imilmente &longs;e la <lb/>bilancia haue&longs;&longs;e il cen­<lb/>tro C &longs;otto la bilancia, <lb/>& &longs;o&longs;&longs;e la trutina CD <lb/>&longs;opra la bilancia, et &longs;i mo <lb/>ue&longs;&longs;e la bilancia in EF,<emph.end type="italics"/> <arrow.to.target n="note118"></arrow.to.target><lb/><emph type="italics"/>egli è manife&longs;to, che la bi <lb/>lancia &longs;i mouerà in giu <lb/>dalla parte di F, &longs;tan­<lb/>do la trutina &longs;opra la bi­<lb/>lancia. </s> <s>& in qual &longs;i vo­<lb/>glia altro &longs;ito che &longs;ia la <lb/>trutina, auerrà &longs;empre il <lb/>mede&longs;imo. </s> <s>Adunque <expan abbr="nõ">non</expan> <lb/>è la trutina, ma il centro <lb/>della bilancia cagione di <lb/>cotali diuer&longs;i effetti.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note118"></margin.target><emph type="italics"/>Per la terz<gap/> di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Egli è pero d'auertire in que&longs;ta parte che con di&longs;&longs;icultà &longs;i puote lauora re vna bilancia <lb/>materiale, che in vno punto &longs;olamente &longs;ia &longs;o&longs;tenuta, &longs;i come con la mente la imagi­<lb/>niamo, & habbia le braccia dal centro co&longs;i eguali non &longs;olamente in lunghezza, ma <lb/>in larghezza, & in profundità, ò gro&longs;&longs;ezza, che tutte le parti di quà, & di là pe&longs;i­<lb/>no a punto egualmente. </s> <s>percio che la materia di&longs;&longs;icili&longs;&longs;imam ente pati&longs;ce cotale giu­<lb/>&longs;ta mi&longs;ura. </s> <s>Per laqual co&longs;a &longs;e con&longs;idereremo il centro e&longs;&longs;ere in e&longs;&longs;a bilancia, non bi­<lb/>&longs;ogna ricorrere al &longs;en&longs;o, concio&longs;ia, che le co&longs;e artificiate non &longs;i po&longs;&longs;ano ridurre a quel <lb/>fommo grado di per&longs;ettione. </s> <s>Ma nelle altre co&longs;e la e&longs;perienza veramente potrà in&longs;e <lb/>gnare le co&longs;e che appaiono percioche <expan abbr="quãtunque">quantunque</expan> ilcentro della <expan abbr="bilãcia">bilancia</expan> &longs;empre &longs;ia vn <lb/>punto, nondimeno quando egli &longs;arà &longs;opra la bilancia, poco importa, &longs;e ben la bilancia <lb/>non &longs;ara &longs;o&longs;tenuta in quel punto co&longs;i puntalmente però che per e&longs;&longs;ere &longs;empre &longs;opra la <lb/>bilancia auerrà &longs;empre il mede&longs;imo. </s> <s>Con &longs;imile modo, quando egli anco è &longs;otto la bi­<lb/>lancia, ilche tuttauia non accade stando il centro in e&longs;&longs;a bilancia, per che &longs;e egli non <lb/>&longs;arà &longs;o&longs;tenuto &longs;empre in quel mezo accuratamente, &longs;ara differenza, e&longs;&longs;endo co&longs;a faci <lb/>li&longs;&longs;ima, che quel centro, muti il proprio &longs;ito, mentre &longs;i moue la bilancia.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Ma che Ari&longs;totele habbia <lb/>propo&longs;to due que&longs;tioni &longs;o <lb/>lamente, cioè perche la <lb/>trutina &longs;tando &longs;opra, &longs;e <lb/>la bilancia <expan abbr="nõ">non</expan> &longs;arà egual <lb/>mente di&longs;tante dall'ori­<lb/>zonte in equilibrio, cioè <lb/>egualmente di&longs;tante dal <lb/>orizonte ritorna, ma &longs;e la <lb/>trutina &longs;ara po&longs;ta &longs;otto <lb/>non ritorna, ma di piu &longs;i <lb/>moue <expan abbr="&longs;ecõdo">&longs;econdo</expan> la parte ba&longs; <lb/>&longs;a: egli è verò per certo. </s> <s><lb/>Ma non già per que&longs;to le <lb/>dimo&longs;trationi &longs;ue &longs;ono <lb/>&longs;ondate nell'angolo mag <lb/>giore, ò minore, & nella <lb/>giacitura della trutina, <lb/>come e&longs;&longs;i dicono: per cio­<lb/>che in questo non com­<lb/>prendono la <expan abbr="m&etilde;te">mente</expan> del filo <lb/>&longs;ofo, che a&longs;&longs;egna la ragio <lb/>ne de gli effetti diuer&longs;i <lb/>de'mouimenti della bilan <lb/>cia. </s> <s>peroche tanto è lon­<lb/>tano, che il filo&longs;o&longs;o attri <lb/>bui&longs;ca que&longs;ti diuer&longs;i effet<emph.end type="italics"/><lb/> <arrow.to.target n="fig37"></arrow.to.target><lb/><emph type="italics"/>ti à gli angoli, che piu to&longs;to dica e&longs;&longs;ere cagione l'ecce&longs;&longs;o, & quel &longs;opra più della gran <lb/>dezza che è dal perpendicolo dell'uno delle braccia della bilancia hor dall'una parte, <lb/>hora dall'altra.<emph.end type="italics"/></s> </p> <figure id="fig37"></figure> <p type="main"> <s><emph type="italics"/>Come stando la trutina &longs;opra in CF, il perpendicolo &longs;arà FCG, il quale &longs;em­<lb/>pre inchina, &longs;econdo lui, ver&longs;o il centro del mondo, il quale anco diuide la bilancia mo&longs; <lb/>&longs;a in DE in parti di&longs;uguali: & la parte maggiore è ver&longs;o il D, & quel che è piu, <lb/>inchina in giu. </s> <s>Adunque dalla parte di D la bilancia &longs;i mouerà in giu fin che ri­<lb/>torni in AB. </s> <s>Ma &longs;e la trutina &longs;arà in CG di &longs;otto, &longs;arà GCF il perpendico­<lb/>lo, ilquale diuiderà parimente la bilancia DE in parte di&longs;uguali, & la parte mag <lb/>giore &longs;arà ver&longs;o E; Per laqual co&longs;a la bilancia &longs;i mouerà in giu dalla parte di </s> <s>/>& accioche que&longs;to &longs;ia dirittamente compre&longs;o, &longs;appia&longs;i, che quando la trutina è &longs;o­<lb/>pra la bilancia, &longs;i ha da intendere, che anche il centro della bilancia &longs;ia &longs;opra la bi­<lb/>lancia, & &longs;e di &longs;otto, anche il centro deue &longs;tare di &longs;otto, come piu a ba&longs;&longs;o manife&longs;te­<lb/>ra&longs;&longs;i, Altramente la dimo&longs;tratione di Ari&longs;totele non conchiuderebbe nulla, pero <lb/>che stando il centro in e&longs;&longs;a bilancia, come in C moua&longs;i la bilancia in qual &longs;i voglia<emph.end type="italics"/> <pb pagenum="21"/><emph type="italics"/>modo, il perpendicolo FG non diuiderà giamai la bilancia &longs;e non nel punto C, et <lb/>in parti eguali. </s> <s>Onde la &longs;entenza di Ari&longs;totele non &longs;olamente non gli &longs;auori&longs;ce, ma <lb/>gli fa anche grandi&longs;sima <lb/>mente contra. </s> <s>il che <lb/>non &longs;olamente è chiaro <lb/>dalla &longs;econda & terza <lb/>propo&longs;itione di que&longs;to li <lb/>bro, ma anco percioche <lb/>&longs;tando il centro &longs;opra <lb/>la bilancia, il pe&longs;o alzato <lb/>acqui&longs;ta grauezza mag <lb/>giore per cau&longs;a del &longs;ito. </s> <s><lb/>Dalla qual co&longs;a accade il <lb/>ritorno della bilancia ad <lb/>eguale di&longs;tanza dall'ori­<lb/>zonte. </s> <s>Ma per lo con­<lb/>trario auiene quando il <lb/>centro è &longs;otto la bilan­<lb/>cia. </s> <s>Le quali co&longs;e tutte <lb/>&longs;i dimo&longs;treranno in que­<lb/>&longs;ta maniera, pre&longs;uppo­<lb/>nendo le co&longs;e, che di &longs;o-<emph.end type="italics"/><lb/> <arrow.to.target n="fig38"></arrow.to.target><lb/><emph type="italics"/>pra furono dechiarate, cioè il pe&longs;o &longs;ar&longs;i più graue da quelloco dal quale &longs;cende piu <lb/>dirittamente, & da quello che egli &longs;ale piu dirittamente far&longs;i parimente piu <lb/>graue.<emph.end type="italics"/></s> </p> <pb/> <figure id="fig38"></figure> <p type="main"> <s><emph type="italics"/>Sia la bilancia AB egualmente di&longs;tante dall'orizonte, il cui centro C &longs;ia &longs;opra la <lb/>bilancia, & &longs;ia il perpendicolo CD: & &longs;iano i centri della grauezza di pe&longs;i eguali <lb/>po&longs;ti in AB: & la bilancia &longs;ia mo&longs;&longs;a in EF. </s> <s>Dico, che il pe&longs;o posto in E ha <lb/>grauezzamaggiore, che il <lb/>pe&longs;o posto in F. & per­<lb/>ciò la bilancia EF e&longs;&longs;e­<lb/>re per ritornare in A</s> <s>/>&longs;ia allungata prima la linea <lb/>CD fin'al centro del mon <lb/>do, che &longs;ia S. </s> <s>Dapoi &longs;ia­<lb/>no congiunte le linee AC, <lb/>CB, EC, CF, HS; <lb/>& dai punti EF &longs;iano ti­<lb/>rate le linee EKGFL egual <lb/><expan abbr="m&etilde;te">mente</expan> di&longs;tanti da HS. </s> <s>Per­<lb/>cioche dunque la di&longs;ce&longs;a na <lb/>turale diritta di tutta la <lb/>grandezza, cioè della bilan <lb/>cia EF co&longs;i di&longs;po&longs;ta in&longs;ie <lb/>me co'pe&longs;i è &longs;econdo la gra­<lb/>uezza del centro H per la <lb/>dirittalinea HS; &longs;arà pa <lb/><expan abbr="rim&etilde;te">rimente</expan> la di&longs;ce&longs;a de'pe&longs;ime&longs; <lb/>&longs;i in EF co&longs;i di&longs;po&longs;ti &longs;econ <lb/>do le linee diritte E<emph.end type="italics"/>K <lb/><emph type="italics"/>FL egualmente distanti <lb/>da HS, &longs;i come di &longs;opra <lb/>habbiamo dimo&longs;trato. </s> <s>La <lb/>di&longs;ce&longs;a dunque, & la &longs;ali­<lb/>ta de i pe&longs;i po&longs;ti in EF &longs;i <lb/>dirà più, & meno obliqua <lb/>&longs;econdo la vicinanza, ò lon <lb/>tananza diputata &longs;econdo<emph.end type="italics"/><lb/> <arrow.to.target n="note119"></arrow.to.target> <emph type="italics"/>le linee EK FL. </s> <s>& per­<lb/>cioche li due lati AD DC <lb/>&longs;ono eguali a i due lati BD<emph.end type="italics"/><lb/> <arrow.to.target n="fig39"></arrow.to.target><lb/><emph type="italics"/>DC; & gli angoli al D &longs;ono retti, &longs;arà il lato AC eguale al lato CB. & e&longs;­<lb/>&longs;endo il punto C immobile; mentre, che i punti AB &longs;imoueranno, de &longs;criueran­<lb/>no la circonferenza di vno cerchio, il cui mezo diametro &longs;arà AC. </s> <s>Per laqual co<emph.end type="italics"/><lb/> <arrow.to.target n="note120"></arrow.to.target> <emph type="italics"/>&longs;a co'l centro C &longs;ia de&longs;critto il cerchio AE BF, i punti AB E<emph.end type="italics"/>F <emph type="italics"/>&longs;aranno nel <lb/>la circonferenza del cerchio. </s> <s>ma e&longs;&longs;endo EF eguale ad AB, &longs;arà la circonfe­<lb/>renza EAF eguale alla circonferenza AFB. </s> <s>Onde tolta via la comune AF<emph.end type="italics"/> <pb pagenum="22"/><emph type="italics"/>&longs;arà la circonferenza EA eguale alla cir conferenza FB. </s> <s>Hor percioche l'ango­<lb/>lo mi&longs;to CEA è eguale al mi&longs;to CFB, & HFB è maggiore di CFB, &<emph.end type="italics"/> <arrow.to.target n="note121"></arrow.to.target><lb/><emph type="italics"/>l'angolo HEA è minore di CEA; &longs;arà l'angolo HFB maggiore dell'angolo <lb/>HEA. </s> <s>Da quali &longs;e &longs;aranno leuati via gli angoli HFG HEK eguali, &longs;arà l'an <lb/>golo GFB maggiore dell'angolo KEA. </s> <s>Adunque la di&longs;ce&longs;a del pe&longs;o po&longs;to in <lb/>E &longs;arà meno obliqua della &longs;alita del pe&longs;o po&longs;to in F. </s> <s>& <expan abbr="quātunque">quantunque</expan> il pe&longs;o po&longs;to in E <lb/>de&longs;cendendo, & il pe&longs;o po&longs;to in F &longs;alendo &longs;i mouino per eguali circonferenze, nondi <lb/>meno percioche il pe&longs;o po&longs;to in E da que&longs;to luogo di&longs;cende piu dirittamente di quel <lb/>che il pe&longs;o F <expan abbr="a&longs;c&etilde;de">a&longs;cende</expan>:pero la naturale po&longs;&longs;anza del pe&longs;o po&longs;to in E &longs;upererà la <expan abbr="re&longs;i&longs;t&etilde;">re&longs;i&longs;tem</expan> <lb/>za della violentia del pe&longs;o F. </s> <s>Onde grauezza maggiore hauerà il pe&longs;o posto in E, <lb/>che il pe&longs;o po&longs;to in F. </s> <s>Adunque il pe&longs;o po&longs;to in E &longs;i mouerà in giù & il pe&longs;o po&longs;to <lb/>in F in sù, fin che la bilancia EF ritorni in AB, che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note119"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note120"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 28. <emph type="italics"/>del terzo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note121"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> <gap/>. <emph type="italics"/>del printo.<emph.end type="italics"/></s> </p> <figure id="fig39"></figure> <p type="main"> <s><emph type="italics"/>La ragione di que&longs;to effetto po&longs;ta da Ari&longs;totele qui &longs;i puote vedere manife&longs;ta. </s> <s>Percio-<emph.end type="italics"/> <arrow.to.target n="note122"></arrow.to.target><lb/><emph type="italics"/>che &longs;ia il punto N doue le linee CS EF &longs;i tagliano in&longs;ieme. </s> <s>& percioche HE <lb/>è eguale ad HF; &longs;arà NE maggiore di NF. </s> <s>adunque la linea CS, che no­<lb/>ma perpendicolo, diuiderà la bilancia EF in parti di&longs;uguali. </s> <s>concio&longs;ia dunque, che <lb/>la parte della bilancia NE &longs;ia maggiore della NF, & quel che è di più bi&longs;o­<lb/>gni, che &longs;ia portato in giù, la bilancia EF dalla parte di E &longs;i mouerà in giu finche <lb/>ritorni in AB.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note122"></margin.target><emph type="italics"/>Ragione de Aristotele.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Oltre à cio da quelle co&longs;e, che <lb/>fin hora &longs;ono &longs;tate dette, <lb/>&longs;i puote affermare, la bilan <lb/>cia EF da quel &longs;ito mo­<lb/>uer&longs;i piu velocemente in <lb/>AB; d'onde la linea EF <lb/>allungata a dirittura per­<lb/>uenga nel centro del mon­<lb/>do. </s> <s>come &longs;ia EFS vna <lb/>linea diritta. </s> <s>& percioche <lb/>CD CK &longs;ono tra loro <lb/>eguali. </s> <s>&longs;e dunque col cen­<lb/>tro C, & con lo &longs;patio <lb/>CD &longs;i de&longs;criuerà il cerchio <lb/>DHM, &longs;aranno i punti <lb/>DH nella circonferenza <lb/>del cerchio. </s> <s>Ma perche la <lb/>CH è à piombo di EF, <lb/>toccherà la EHS il cer­<lb/>chio DHM nel punto <lb/>H. </s> <s>il pe&longs;o dunque po&longs;to in <lb/>H, (&longs;i come di &longs;opra hab <lb/>biamo prouato) &longs;arà piu<emph.end type="italics"/><lb/> <arrow.to.target n="fig40"></arrow.to.target> <pb/><emph type="italics"/>graue che in verun altro &longs;ito del cerchio DHM. </s> <s>Adunque la grandezza fatta de' <lb/>pe&longs;i EF, & della bilancia EF, il cui centro della grauczza sta in H, in cote&longs;to <lb/>&longs;ito grauerà più, che in qual &longs;i voglia altro &longs;ito del cerchio &longs;i troui il punto H. </s> <s>Da <lb/>que&longs;to &longs;ito adunque &longs;i mouera piu velocemente che da qualunque altro. </s> <s>& &longs;e lo H <lb/>&longs;arà piu da pre&longs;&longs;o al D <lb/>manco grauerà, & me­<lb/>no &longs;i mouerà da quel &longs;ito; <lb/>peroche &longs;empreè piu torta <lb/>la &longs;ce&longs;a, & meno diritta. </s> <s><lb/>La bilancia dunque EF <lb/>&longs;i mouerà più velocemen­<lb/>te da que&longs;to &longs;ito, che da <lb/>altro &longs;ito, & &longs;e piu dapre&longs; <lb/>&longs;o acco&longs;teraßi ad AB, <lb/>d'indi &longs;i mouerà meno poi <lb/>quanto piu da lunge &longs;arà <lb/>di&longs;tante il punto H dal <lb/>punto C &longs;i mouerà più ve <lb/>locemente, il che non &longs;olo <lb/>da Ari&longs;totele nel principio <lb/>delle que&longs;tioni mecaniche, <lb/>& dai detti di &longs;opra è ma <lb/>nife&longs;to, ma ancora da quel <lb/>le co&longs;e, che di &longs;otto nella <lb/>&longs;e&longs;ta propo&longs;itione &longs;iamo <lb/>per dire, apparerà chiaro. </s> <s><lb/>La bilancia dunque EF <lb/>quanto più &longs;arà lontana <lb/>dal &longs;uo centro, &longs;i mouerà anche piu velocemente.<emph.end type="italics"/></s> </p> <figure id="fig40"></figure> <figure></figure> <pb pagenum="23"/> <p type="main"> <s><emph type="italics"/>Sia poi labilancia AB, il cui centro C stia &longs;otto la bilancia, & &longs;iano in AB <lb/>pe&longs;i eguali, & &longs;ia mo&longs;&longs;a la bilancia in EF. </s> <s>Dico che il pe&longs;o ha grauezza maggio­<lb/>re in F, che in E. & <lb/>perciò la bilancia EF <lb/>e&longs;&longs;ere per mouer&longs;i in giù <lb/>dalla parte di F. </s> <s>&longs;ia allun <lb/>gata la linea DC dall'una <lb/>parte, & dall'altra fin <lb/>nel centro del mondo S, <lb/>& fin ad O, & &longs;ia tira <lb/>tala linea HS, alla qua <lb/>le dai punti EF &longs;iano ti <lb/>rate le linee GEK FL <lb/>egualmente di&longs;tanti, & <lb/>&longs;iano congiunte le CE <lb/>CF: & dal centro C <expan abbr="cõ">com</expan> <lb/>lo &longs;patio CE de&longs;criua&longs;i <lb/>il cerchio AEO B</s> <s>/>&longs;i dimo&longs;trerà &longs;imilmente <lb/>i punti AB EF e&longs;&longs;e­<lb/>re nella circonferenza del <lb/>cerchio, & che la di&longs;ce&longs;a <lb/>della bilancia EF in&longs;ie­<lb/>me co'pe&longs;i &longs;i fà diritta &longs;e <lb/>condo la linea HS: & <lb/>de ipe&longs;i po&longs;ti in EF &longs;e­<lb/>condo le linee GK FL <lb/>egualmente di&longs;tanti da <lb/>HS. </s> <s>Et percioche l'ango <lb/>lo CFP è eguale all'an <lb/>golo CEO &longs;arà l'ango­<lb/>lo HFP maggiore del­<lb/>l'angolo HEO. </s> <s>ma l'an<emph.end type="italics"/> <arrow.to.target n="note123"></arrow.to.target><lb/><emph type="italics"/>golo HFL è eguale al­<lb/>l'angolo HEG. </s> <s>Da qua <lb/>li &longs;e &longs;aranno leuati via <lb/>gli angoli HFP HEO,<emph.end type="italics"/><lb/> <arrow.to.target n="fig41"></arrow.to.target><lb/><emph type="italics"/>&longs;arà l'angolo LFP minore dell' angolo GEO. </s> <s>Per laqual co&longs;a la &longs;ce&longs;a del pe&longs;o <lb/>po&longs;to in F &longs;arà piu diritta della a&longs;ce&longs;a del pe&longs;o po&longs;to in E. </s> <s>Adunque la po&longs;&longs;anza <lb/>naturale del pe&longs;o po&longs;to in F &longs;upererà la re&longs;i&longs;tenza della violentia del pe&longs;o po&longs;to in <lb/>E. & percio hauerà maggior grauezza il pe&longs;o di F, che il pe&longs;o di E. </s> <s>Adunque <lb/>il pe&longs;o di F &longs;imouer à in giù, & il pe&longs;o di E &longs;i mouerà in sù.<emph.end type="italics"/></s> </p> <pb/> <p type="margin"> <s><margin.target id="note123"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure id="fig41"></figure> <p type="main"> <s> <arrow.to.target n="note124"></arrow.to.target><emph type="italics"/>La ragione di Ari&longs;totele parimente qui è cbiara. </s> <s>Percioche &longs;ia il punto N doue le <lb/>linee CO EF &longs;i tagliano in&longs;ieme. </s> <s>&longs;arà la NF maggiore della NE. & perche <lb/>il perpendicolo CO, &longs;e­<lb/>condo lui, diuide in parti <lb/>di&longs;uguali la bilancia, & <lb/>la parte maggiore è ver&longs;o <lb/>F, cioè NF; la bilan­<lb/>cia EF &longs;i mouerà in giù <lb/>dalla parte di F, concio <lb/>&longs;ia che quel che è di piu <lb/>venga portato à ba&longs;&longs;o.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note124"></margin.target><emph type="italics"/>Ragione di Aristotele.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Similmente dalle co&longs;e dette <lb/>caueremo, che <expan abbr="quāto">quanto</expan> piu <lb/>la bilancia EF tenente <lb/>il centro &longs;otto la bilancia, <lb/>&longs;arà <expan abbr="lõtana">lontana</expan> dal &longs;ito AB <lb/>&longs;i mouerà piu velocemen <lb/>te, percioche il centro del <lb/>la grauezza H, quanto <lb/>piu è di&longs;tante dal punto <lb/>D, tanto piu velocemen <lb/>te il pe&longs;o compo&longs;to de<gap/> pe <lb/>&longs;i EF, & della bilancia <lb/>EF &longs;i mouerà, finche <lb/>l'angolo CHS diuenga <lb/>retto. </s> <s>& dauantaggio &longs;i <lb/>mouerà anche piu veloce <lb/>mente quanto la bilancia <lb/>&longs;arà piu lontana dal cen­<lb/>tro C.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Oltre à ciò ne piace dalle &longs;ue <lb/>ragioni, & fal&longs;e pre&longs;uppo <lb/>&longs;te manife&longs;tare, & pro <lb/>durre gli effetti, & i moti <lb/>già dichiarati della bilan <lb/>cia, affine che appaia <expan abbr="quā">quam</expan> <lb/>ta &longs;ia la efficacia della ve­<lb/>rità, come quella, che dalle co&longs;e fal&longs;e ancora &longs;i sforza di ri&longs;plendere.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Pongan&longs;i le co&longs;e iste&longs;&longs;e, cioè &longs;iail cerchio AE BF, & la bilancia AB, il cui cen­<lb/>tro C &longs;ia &longs;opra la bilancia, moua&longs;i in EF. </s> <s>Dico che il pe&longs;o po&longs;to in E hà iui <lb/>grauezza maggiore, che il pe&longs;o po&longs;to in F; & che la <expan abbr="bilācia">bilancia</expan> EF ritornerà in AB<emph.end type="italics"/> <pb pagenum="42"/><emph type="italics"/>&longs;iano tirate dai punti EF le linee EL FM à piombo di AB, le quali &longs;aran-<emph.end type="italics"/> <arrow.to.target n="note125"></arrow.to.target><lb/><emph type="italics"/>no tra loro egualmente di&longs;tanti, & &longs;ia il punto N doue la AB, & la EF &longs;i <lb/>tagliano fra loro. </s> <s>Percioche dunque l'angolo FNM è eguale all'angolo ENL,<emph.end type="italics"/> <arrow.to.target n="note126"></arrow.to.target><lb/><emph type="italics"/>& l'angolo FMN ret <lb/>to è eguale ad ELN <lb/>retto, & il re&longs;tante <lb/>NFM al re&longs;tante<emph.end type="italics"/> <arrow.to.target n="note127"></arrow.to.target><lb/><emph type="italics"/>NEL è etiandio egua­<lb/>le; &longs;arà il triangolo NLE <lb/>&longs;imile al triangolo NMF.<emph.end type="italics"/> <arrow.to.target n="note128"></arrow.to.target><lb/><emph type="italics"/>Si come dunque è la NE <lb/>ver&longs;o la EL, co&longs;i NF<emph.end type="italics"/> <arrow.to.target n="note129"></arrow.to.target><lb/><emph type="italics"/>ad FM; & permut an­<lb/>do, &longs;i come EN ad NF, <lb/>co&longs;i EL ad FM. </s> <s>Ma <lb/>e&longs;&longs;endo HE eguale ad <lb/>HF, &longs;arà EN mag­<lb/>gior di NF. </s> <s>Per laqual <lb/>co&longs;a anco EL &longs;arà mag<emph.end type="italics"/><lb/> <arrow.to.target n="fig42"></arrow.to.target><lb/><emph type="italics"/>giore <gap/>i FM. </s> <s>& percioche mentre il pe&longs;o po&longs;to in E de&longs;cende per la circonferen­<lb/>za EA, il pe&longs;o po&longs;to in F &longs;ale per la circon&longs;erenza FB eguale alla circonferen­<lb/>za EA, & la di&longs;ce&longs;a del pe&longs;o po&longs;to in E piglia (come e&longs;&longs;i dicono) di diretto EL: <lb/>& la &longs;alita del pe&longs;o po&longs;to in F piglia di diretto FM, meno di diretto verrà a pi­<lb/>gliare la &longs;alita del pe&longs;o po&longs;to in F, che la di&longs;ce&longs;a del pe&longs;o po&longs;to in E. </s> <s>Dunque il pe <lb/>&longs;o po&longs;to in E haurà grauezza maggiore, che il pe&longs;o po&longs;to in F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note125"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 28. <emph type="italics"/>d l primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note126"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note127"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note128"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4.<emph type="italics"/>del &longs;esto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note129"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure id="fig42"></figure> <p type="main"> <s><emph type="italics"/>Sia allungata la linea CD dall una parte, & dall'altra in OP, laquale tagli la linea <lb/>EF nel punto S. </s> <s>& percioche (come dicono) quanto piu è lontano il pe&longs;o dalla <lb/>linea della direttione OP, tanto &longs;i fa piu graue; però con que&longs;to mezo ancora pro­<lb/>uera&longs;&longs;i il pe&longs;o po&longs;to in E hauer grauezza maggiore del pe&longs;o po&longs;to in F. </s> <s>Siano dai <lb/>punti EF tirate le linee EQ FR a piombo di OP. </s> <s>Con &longs;imile ragione mo&longs;tre <lb/>ra&longs;&longs;i, che il triangolo QES è &longs;imile al triangolo RFS; & che la linea EQ è <lb/>maggiore di RF. & co&longs;i il pe&longs;o po&longs;to in E &longs;arà piu lontano dalla linea OP, che <lb/>il pe&longs;o po&longs;to in F; & per ciò il pe&longs;o po&longs;to in E hauerà grauezza maggiore del pe <lb/>&longs;o po&longs;to in<emph.end type="italics"/> F. <emph type="italics"/>Dallequali co&longs;e appare euidente il ritorno della bilancia E<emph.end type="italics"/>F <emph type="italics"/>in AB.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Ma &longs;e il centro della bilancia &longs;arà &longs;otto la bilancia, allhora &longs;i mo&longs;trerà con gli i&longs;te&longs;&longs;i me <lb/>zi, che il pe&longs;o abba&longs;&longs;ato hauerà grauezza maggiore dall'alzato. </s> <s>&longs;iano tirate dapun­<lb/>ti EF le linee EL FM <lb/>a piombo di AB. </s> <s>&longs;imil <lb/>mente&longs;i prouerà EL e&longs; <lb/>&longs;ere maggiore di FM; et <lb/>perciò la &longs;ce&longs;a del pe&longs;o po <lb/>sto in F prenderà meno <lb/>di dirittura, che la &longs;alita <lb/>del pe&longs;o po&longs;to in E. </s> <s>On­<lb/>de la re&longs;i&longs;tenza della vio­<lb/>lentia del pe&longs;o po&longs;to in E <lb/>&longs;upererà la naturale incli­<lb/>natione del pe&longs;o po&longs;to in <lb/>F. </s> <s>Adunque il pe&longs;o po&longs;to <lb/>in E &longs;arà piu graue del <lb/>pe&longs;o posto in F.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Sia <expan abbr="allūgata">allungata</expan> etiandio la CD <lb/>dall'una parte & l'altra<emph.end type="italics"/><lb/> <arrow.to.target n="fig43"></arrow.to.target><lb/><emph type="italics"/>in OP, & &longs;iano tirate dai punti EF le linee EQ FR à piombo dilei. </s> <s>&longs;i pro <lb/>verà con l'i&longs;te&longs;&longs;o modo in tutto, che la linea EQ è maggiore di FR. & percio il <lb/>pe&longs;o po&longs;to in E &longs;arà piu lontano dalla linea della dirittura OP, che il pe&longs;o po&longs;to <lb/>in F. </s> <s>Adunque il pe&longs;o po&longs;to in E haurà grauezza maggiore del pe&longs;o po&longs;to in F. </s> <s><lb/>Dalle quali co&longs;e &longs;egue, che la bilancia EF &longs;i moue in giù dalla parte di E.<emph.end type="italics"/></s> </p> <figure id="fig43"></figure> <p type="main"> <s><emph type="italics"/>Si che Aristotele propo&longs;e que&longs;te due que&longs;tioni &longs;olamente, & la&longs;ciò la terza, cioè quando <lb/>il centro della bilancia &longs;tà nella bilancia i&longs;te&longs;&longs;a. </s> <s>Que&longs;ta però trala&longs;ciò egli, co­<lb/>me nota, &longs;i come egli &longs;ole trala&longs;ciare le co&longs;e molto note. </s> <s>Imperoche à chi puote <lb/>far dubbio, che &longs;e il pe&longs;o &longs;arà &longs;o&longs;tentato nel centro della grauezza&longs;ua, che non i&longs;tia <lb/>fermo? </s> <s>Mapotrebbe for&longs;e alcuno riprendere quelle co&longs;e che per &longs;ua &longs;ententia hab­<lb/>biamo propo&longs;to, affermando noi non hauere prodotto in mezo tutta la intera &longs;enten <lb/>za &longs;ua. </s> <s>Imperoche proponendo egli nella &longs;econda parte della que&longs;tione &longs;econda. <lb/>“Perche la bilancia e&longs;&longs;endo posta la trutina di &longs;otto, quando, portato il pe&longs;o in giu, al <lb/>cuno lo rimoue, non a&longs;cende, ma rimane?” non afferma perciò la bilancia mouer&longs;i in <lb/>giù, ma rimanere, il che pare &longs;imilmente hauere nella vltima conclu&longs;ione raccolto. </s> <s><lb/>Ma que&longs;to non &longs;o larnente non ci &longs;a contra, ma&longs;e egli è ben' inte&longs;o grandi&longs;&longs;imamen­<lb/>te aiuta.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Percioche &longs;ia la bilancia AB egualmente di&longs;tante dall'orizonte, il cui centro E &longs;ia <lb/>&longs;otto la bilancia. </s> <s>& perche Ari&longs;totele con&longs;idera la bilancia come ella è in fatto, però <lb/>egli è nece&longs;&longs;ario collocare la trutina, ouero qualche altra co&longs;a &longs;otto il centro E, co­<lb/>me EF, che in ogni modo &longs;arà trutina, per modo, che &longs;o&longs;tengail centro E. & &longs;ia <lb/>ECD il perpendicolo. </s> <s>& accioche la bilancia AB &longs;i moua da que&longs;to &longs;ito, dice<emph.end type="italics"/> <pb pagenum="25"/><emph type="italics"/>Ari&longs;totele, ponga&longs;i il pe&longs;o in B, ilquale e&longs;&longs;endo graue mouerà la bilancia dalla par­<lb/>te B in giù, come in G, talche per l'impedimento non potrà egli piu mouer&longs;i in <lb/>giu, ma non dice gia Ari&longs;totele, che &longs;i moua la bilancia in giu dalla parte di B fin <lb/>tanto che parerà, da <lb/>poi &longs;i la&longs;ci, come noi <lb/>di cemmo<gap/>ma ordina <lb/>che &longs;ia posto il pe&longs;o <lb/>in B, il quale di &longs;ua <lb/>natura &longs;i mouera <lb/>&longs;empre in giù finche <lb/>la bilancia &longs;i appog­<lb/>gi alla trutina, ouerò <lb/>a qualche altra co&longs;</s> <s>/>& quando il B &longs;a­<lb/>rà nel G, la bilan­<lb/>cia &longs;arà in GH, nel <lb/>qual &longs;ite leuato via <lb/>il pe&longs;o, rimarrà: per <lb/>e&longs;&longs;ere la maggior par <lb/>te della bilancia dal <lb/>perpendicolo uer&longs;o il<emph.end type="italics"/><lb/> <arrow.to.target n="fig44"></arrow.to.target><lb/><emph type="italics"/>G, che è DG, che DH. ne piu mouera&longs;&longs;i in giu, imperoche la bilancia &longs;tar à &longs;opra <lb/>la trutina, ouero qualche altra co&longs;a, che &longs;o&longs;tenga il centro della bilancia. </s> <s>peroche &longs;e a <lb/>cote&longs;ta non &longs;i appoggia&longs;&longs;e, verrebbe la bilancia à mouer&longs;i, &longs;econdo la &longs;ua opinione, <lb/>in giù dalla parte di G, concio&longs;ia, che quello che è di piu, cioè DG debba e&longs;&longs;ere <lb/>per nece&longs;&longs;ità in giu portato.<emph.end type="italics"/></s> </p> <figure id="fig44"></figure> <p type="main"> <s><emph type="italics"/>Ma potrebbe dauantagio dire alcuno, &longs;e in B &longs;arà collocato vn pe&longs;o picciolo, &longs;i mo­<lb/>uerà ben la bilancia in giu, ma non gia fin al G; nel qual &longs;ito, &longs;econdo Aristo­<lb/>tele, leuato via il pe&longs;o, deue remanere. </s> <s>ilche è manife&longs;to per la e&longs;perientia, inchi­<lb/>nando&longs;i la <expan abbr="bilãcia">bilancia</expan> più, & meno, quando in vna e&longs;tremita della bilancia &longs;olamente <lb/>vi è po&longs;to il pe&longs;o, che &longs;ia ò maggiore, ò minore. </s> <s>ilche è veri&longs;&longs;imo allhora che il centro <lb/>è collocato &longs;opra la bilancia, ma non già &longs;otto, ne in e&longs;&longs;a bilancia, come per gratia <lb/>di e&longs;empio.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia la bilancia AB egualmente di&longs;tante dall'orizonte, il cui centro C &longs;ia &longs;opra la bi <lb/>lancia, & il perpendicolo <lb/>CD a piombo dell' ori­<lb/>zonte, il quale da la par­<lb/>te D &longs;ia allungato in H. </s> <s><lb/>Hor percioche con&longs;idera <lb/>ta la grauezza della bi­<lb/>lancia, &longs;arà il punto D <lb/>il centro della grauezza <lb/>della bilancia. </s> <s>&longs;e dunque <lb/>vn piccolo pe&longs;o &longs;arà po­<lb/>&longs;to nel B, il cui centro <lb/>della grauezza &longs;ia nel <expan abbr="pũ">pum</expan> <lb/>to B; gia piu non &longs;arà <lb/>il centro della grauezza <lb/>D della magnitudine<emph.end type="italics"/><lb/> <arrow.to.target n="note130"></arrow.to.target> <emph type="italics"/>compo&longs;ta della bilancia<emph.end type="italics"/><lb/> <arrow.to.target n="fig45"></arrow.to.target><lb/><emph type="italics"/>AB, & del pe&longs;o po&longs;to in B, ma &longs;arà nella linea DB, come in K: per modo <lb/>che DE ad EB &longs;ia come il pe&longs;o po&longs;to in B alla grauezza della bilancia A</s> <s>/>congiunga&longs;i la CE. </s> <s>& percioche il punto C è immobile, mentre la bilancia &longs;i <lb/>moue, il punto E de&longs;criuerà la circonferenza del cerchio EFG, il cui mezo dia­<lb/>metro è CE, & il centro C. </s> <s>Ma perche CD &longs;tà a piombo dell' orizonte, la li <lb/>nea CE non &longs;arà gia ella à piombo dell' orizonte. </s> <s>Per laqual co&longs;a la grandez­<lb/>za composta di AB, & del pe&longs;o po&longs;to in B non rimarrà in questo &longs;ito; ma &longs;i <lb/>mouerà in giu &longs;econdo il centro E della &longs;ua grauezza per la circonferenza EFG,<emph.end type="italics"/><lb/> <arrow.to.target n="note131"></arrow.to.target> <emph type="italics"/>finche CE diuenti a piombo dell' orizonte, cioè finche la CE peruenga in CD</s> <s>/>& allhora la bilancia AB &longs;arà mo&longs;&longs;ain KL, nel qual &longs;ito la bilancia rimarrà <lb/>in&longs;ieme co'l pe&longs;o, ne d'auantaggio &longs;i mouerà in giù. </s> <s>che &longs;e in B &longs;arà po&longs;to vn pe&longs;o <lb/>piu graue, il centro'della grauezza di tutta la magnitudine &longs;arà piu dappre&longs;&longs;o al B, <lb/>come in M. & allhora la bilancia &longs;i mouerà in giu, finche la congiunta linea CM <lb/>peruenga nella linea CDH. </s> <s>Dal por&longs;i dunque pe&longs;o maggiore ò minore in B, la <lb/>bilancia &longs;i inchinerà piu ò meno. </s> <s>Da che &longs;egue che il pe&longs;o B de&longs;criuerà &longs;empre vna <lb/>circonferenza minore della quarta parte d'un cerchio, per e&longs;&longs;ere l'angolo FCE &longs;em <lb/>pre acuto:ne il punto B peruenirà gia mai fin alla linea CH, percioche &longs;empre il <lb/>centro della grauezza del pe&longs;o, & dalla bilancia in&longs;ieme &longs;arà fra BD. </s> <s>tuttauia <expan abbr="quā">quam</expan> <lb/>to &longs;arà il pe&longs;o po&longs;to in B piu graue, de&longs;criuerà anche circonferenza maggiore, ve­<lb/>nendo&longs;i per que&longs;to il punto B ad acco&longs;tare piu alla linea CH.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note130"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del primo. </s> <s>di. </s> <s>drch. </s> <s>del le co&longs;e cgual <expan abbr="m&etilde;se">mense</expan> <expan abbr="pesāti">pesanti</expan>.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note131"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig45"></figure> <p type="main"> <s><emph type="italics"/>Mi habbia la bilancia AB il centro C nella i&longs;te&longs;&longs;a bilancia, & nel &longs;uo mezo, <lb/>&longs;arà il C centro ancora della grauezza della bilancia, dal quale &longs;ia tirata la li­<lb/>nea FCG a piombo di e&longs;&longs;a AB, & dell' orizonte. </s> <s>Ponga&longs;i dapoi in B qual <lb/>pe&longs;o &longs;i voglia; &longs;arà il centro di tutta la grauezza, come in E; &longs;i fattamente che <lb/>la CE ver&longs;o EB &longs;ia come il pe&longs;o po&longs;to in B alla grauezza della bilancia. </s> <s>& per<emph.end type="italics"/> <pb pagenum="26"/><emph type="italics"/>cioche la CE non è a piombo dell' orizonte, la bilancia AB, & il pe&longs;o po&longs;to in <lb/>B non rimaranno in que­<lb/>&longs;to &longs;ito gia mai; ma &longs;i mo­<lb/>ueranno in giu dalla par­<lb/>te di B, fin che CE &longs;i <lb/>&longs;accia à piombo dell' ori­<lb/>zonte; cioè fin che la bilan­<lb/>cia AB peruenga in FG. </s> <s><lb/>Onde è chiaro, che cia&longs;cun <lb/>pe&longs;o po&longs;to in B, &longs;empre <lb/>de&longs;criue la quarta parte <lb/>d'un cerchio.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;ia il centro C &longs;otto la bilancia AB, & &longs;ia DCE il perpendicolo. </s> <s>&longs;imilmente <lb/>per e&longs;&longs;er il pe&longs;o posto in B, &longs;arà il centro della grauezza della magnitudine compe <lb/>&longs;ta di AB bilancia, & del pe&longs;o po&longs;to in B nella linea DB, come in F; &longs;i <expan abbr="fattamē">fattamen</expan> <lb/>te che come DF &longs;i ha ver&longs;o FB co&longs;i &longs;ia il pe&longs;o po&longs;to in B al pe&longs;o della bilan­<lb/>cia. </s> <s>congiunga&longs;i CF. & <lb/>percioche CD è a piombo <lb/>dell' orizonte, non &longs;arà gia <lb/>la linea CF a piombo del <lb/>l'orizonte. </s> <s>Per laqual co&longs;a <lb/>la magnitudine compo&longs;ta <lb/>della bilancia AB, & del <lb/>pe&longs;o po&longs;to in B in que&longs;to <lb/>&longs;ito non &longs;tarà mai ferma; <lb/>ma in giu mouera&longs;&longs;i &longs;e alcu <lb/>na co&longs;a non la impedi&longs;ce, <lb/>finche CF peruenga in <lb/>DCE, nel qual &longs;ito la bi­<lb/>lancia rimarrà in&longs;ieme co'l<emph.end type="italics"/><lb/> <arrow.to.target n="fig46"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o. </s> <s>& il punto B &longs;arà come in G, & il punto A in H, & la bilancia GH <lb/>non hauerà piu il centro di &longs;otto, ma &longs;opra e&longs;&longs;a. </s> <s>La qual co&longs;a hauerà &longs;empre, quan­<lb/>tunque &longs;i ponga vn minimo pe&longs;o in B. </s> <s>Auanti che dunque il B peruenga al G, <lb/>egli è nece&longs;&longs;ario, che la bilancia incontri la trutina po&longs;ta di &longs;otto, ouero alcuna altra <lb/>co&longs;a, che &longs;o&longs;tenti il centro C, & iui s'appoggi. </s> <s>Da que&longs;to &longs;egue, che il pe&longs;o B &longs;em <lb/>pre &longs;i moue oltre la linea DK, & de&longs;criue &longs;empre vna circonferenza maggiore del <lb/>la quarta parte del cerchio, per e&longs;&longs;ere l'angolo FCE &longs;empre ottu&longs;o, & l'angolo <lb/>DCF &longs;empre acuto. </s> <s>& quanto il pe&longs;o posto in B &longs;arà piu leggiero, de&longs;criuerà tut­<lb/>tauia anche circonferenza maggiore. </s> <s>Imperoche quanto il pe&longs;o po&longs;to in G &longs;arà piu <lb/>leggiero, tanto piu il pe&longs;o detto posto in G &longs;i alzerà; & la bilancia GA s'acco&longs;te<emph.end type="italics"/></s> </p> <pb pagenum="27"/> <figure id="fig46"></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;ia il centro della bilancia AB &longs;opra il C in F; & &longs;ia FC à piombo di AB, <lb/>& dell' orizonte: & &longs;e <lb/>la bilancia &longs;arà mo&longs;&longs;a in<emph.end type="italics"/> <arrow.to.target n="note132"></arrow.to.target><lb/><emph type="italics"/>DE, la linea CF &longs;arà <lb/>mo&longs;&longs;a in FG, la quale <lb/>per non e&longs;&longs;ere à piombo <lb/>dell' orizonte, la bilancia <lb/>DE &longs;imouerà in giu dalla <lb/>parte di D, finche FG <lb/>ritorni in FC: & allho <lb/>ra la bilancia DE &longs;arà <lb/>in AB, nel qual &longs;ito an <lb/>che rim<gap/>rà.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note132"></margin.target><emph type="italics"/>Per la prima di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e il centro F della bi­<lb/>lancia &longs;arà &longs;otto la <expan abbr="bilē-cia">bilen­<lb/>cia</expan>, & &longs;iala bilancia mo&longs;<emph.end type="italics"/> <arrow.to.target n="note133"></arrow.to.target><lb/><emph type="italics"/>&longs;a in DE primier amen <lb/>te egli è manife&longs;to che la <lb/>bilancia rimarrà in AB: <lb/>& in DE mouera&longs;&longs;i in <lb/>giu dalla parte di E, per <lb/>non e&longs;&longs;ere la linea FG <lb/>à piombo dell' orizonte.<emph.end type="italics"/><lb/> <arrow.to.target n="fig47"></arrow.to.target><lb/><emph type="italics"/>rà piu pre&longs;&longs;o al &longs;ito egualmente di&longs;tante dall'orizonte. </s> <s>Le quali co&longs;e tutte re&longs;tano ma <lb/>ni&longs;e&longs;te da quelle che di &longs;opra &longs;ono &longs;tate dette.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note133"></margin.target><emph type="italics"/>Per la prima di questo.<emph.end type="italics"/></s> </p> <figure id="fig47"></figure> <p type="main"> <s><emph type="italics"/>Prouate que&longs;te co&longs;e, egli è chia <lb/>ro, che il centro della bilan­<lb/>cia è cagione de gli effetti di <lb/>uer&longs;i della bilancia. </s> <s>& &longs;i ve <lb/>de ancora che tutte le pro­<lb/>po&longs;itioni di Archimede del <lb/>le co&longs;e, che egualmente pe&longs;a <lb/>no, a ciò pertinenti, in ogni <lb/>&longs;ito &longs;ono vere. </s> <s>cioè, &longs;ia pur <lb/>la bilancia di&longs;tante <expan abbr="egualm&etilde;">egualmen</expan> <lb/>te dall'orizonte, ouero non, <lb/>pur che il centro della bilan <lb/>cia &longs;ia collocato in e&longs;&longs;a <expan abbr="bilā">bilam</expan> <lb/>cia, &longs;i come egli la con&longs;ide-<emph.end type="italics"/><lb/> <arrow.to.target n="fig48"></arrow.to.target><lb/><emph type="italics"/>rà. </s> <s>& quantunque la bilancia habbia difuguali le braccia, auerrd tutt auia l'i&longs;te&longs;&longs;o, & <lb/>&longs;i dimo&longs;trerà co'l modo i&longs;te&longs;&longs;o in tutto, che il centro della bilancia collocato in diuer <lb/>&longs;e maniere produrrà vari effetti.<emph.end type="italics"/></s> </p> <figure id="fig48"></figure> <p type="main"> <s><emph type="italics"/>Percioche &longs;ia la bilancia <lb/>AB egualmente di&longs;tan <lb/>te dall'orizonte; & &longs;iano <lb/>in AB pe&longs;i di&longs;uguali, il <lb/>centro della grauezza <lb/>dei quali &longs;ia in C, & <lb/>&longs;ia attacata la bilancia <lb/>nell'i&longs;te&longs;&longs;o punto di C, <lb/>& moua&longs;i la bilancia in<emph.end type="italics"/><lb/> <arrow.to.target n="note134"></arrow.to.target> <emph type="italics"/>DE; egliè manife&longs;to, <lb/>che la bilancia rimarrà <lb/>non &longs;olamente in DE, <lb/>ma in qual &longs;i voglia altre <lb/>&longs;ito.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note134"></margin.target><emph type="italics"/>Per la diffi ni ione del centro della grauezza.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Da que&longs;te co&longs;e co&longs;i terminate, &longs;e la bilancia fo&longs;&longs;e inarcata, ouero, che le braccia della bi <lb/>lancia forma&longs;&longs;ero vn'angolo, & &longs;i di&longs;poneße il centro diuer&longs;amente, (ben che que­<lb/>&longs;ta propriamente non &longs;arebbe bilancia,) potremo nondimeno anche dimo&longs;trare di lei <lb/>vary effetti. </s> <s>Come &longs;ia la bilancia ACB, il cui centro, d'intorno al quale &longs;i volge, <lb/>&longs;i a C, & tiratala linea AB, &longs;ia <lb/>l'arco ouerò l'angolo ACB &longs;opra <lb/>la linea AB; & pongan&longs;i in AB <lb/>icentri della grauezza de'pe&longs;i, i quali <lb/>rimangano in que&longs;to &longs;ito. </s> <s>Moua&longs;i poi <lb/>la <expan abbr="bilācia">bilancia</expan> da que&longs;to &longs;ito, come in ECF. </s> <s><lb/>Dico che la bilancia ECF ritornerà <lb/>in ACB. </s> <s>Ritroui&longs;i il centro della <lb/>grauczza di tutta la magnitudine D, <lb/>& &longs;ia congiunta la CD. </s> <s>Hor percio<emph.end type="italics"/><lb/> <arrow.to.target n="note135"></arrow.to.target> <emph type="italics"/>che i pe&longs;i AB stanno fermi, lali­<lb/>nea CD &longs;arà à piombo dell' orizon-<emph.end type="italics"/><lb/> <arrow.to.target n="fig49"></arrow.to.target><lb/><emph type="italics"/>ie. </s> <s>Quando dunque la bilancia &longs;arà in ECF, la linea CD &longs;arà come in CG; <lb/>la quale per non e&longs;&longs;ere à piombo dell' orizonte, la bilancia ECF ritornerà in <lb/>ACB. ilche parimente auenir à, &longs;e il centro C &longs;arà me&longs;&longs;o &longs;opra la bilancia, co­<lb/>me in H.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note135"></margin.target><emph type="italics"/>Per la prima di questo.<emph.end type="italics"/></s> </p> <figure id="fig49"></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e l'arco, ouero l'angolo ACB <lb/>&longs;arà &longs;otto la linea AB, nel <lb/>modo i&longs;te&longs;&longs;o mo&longs;treremo, la bi­<lb/>lancia ECF, il cui centro &longs;ia <lb/>ouero in C, ouero in H, do­<lb/>uer&longs;i mouere in giu dalla parte <lb/>di F.<emph.end type="italics"/><lb/> <arrow.to.target n="fig50"></arrow.to.target></s> </p> <pb pagenum="28"/> <figure id="fig50"></figure> <figure></figure> <p type="main"> <s><emph type="italics"/>Et &longs;e l'angolo ACB fo&longs;&longs;e &longs;oprala linea AB, & il centro della bilancia H; & <lb/>& lalinea CH &longs;o&longs;tene&longs;&longs;e la bilancia; & &longs;imoue&longs;&longs;e la bilancia in EKF; la bilan <lb/>cia EKF ritornerà in ACB.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma &longs;e il centro della bilancia &longs;arà D, moua&longs;i in qualunque modo la bilancia, doue &longs;i <lb/>la&longs;cierà, iui rimarrà.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Se poi il punto H &longs;arà &longs;otto la linea AB; allhora la bilancia EKF &longs;i mouerà in <lb/>giu dalla parte di F.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et con &longs;imile ragione in tutto, &longs;e l'ango­<lb/>lo ACB &longs;arà &longs;otto la linea AB; <lb/>& &longs;ia il centro della bilancia H, & <lb/>&longs;ia la bilancia &longs;o&longs;tentata dalla linea <lb/>CH; &longs;e la bilancia moueraßi da que&longs;to <lb/>&longs;ito, &longs;i mouerà in giu dalla parte del pe <lb/>&longs;o più ba&longs;&longs;o. </s> <s>& &longs;e il centro della bilan­<lb/>cia &longs;ia D; rimarrà doue &longs;i la&longs;cierà. </s> <s>che <lb/>&longs;e &longs;arà in K; & da cotale &longs;ito &longs;i mo <lb/>uerà, ritornerà ad ogni modo nello i&longs;te&longs; <lb/>&longs;o. </s> <s>Le quali co&longs;e tutte da quel che in<emph.end type="italics"/><lb/> <arrow.to.target n="fig51"></arrow.to.target><lb/><emph type="italics"/>principio dicemmo &longs;ono mani&longs;este. </s> <s>&longs;imilmente &longs;e il centro della bilancia &longs;arà po&longs;to <lb/>in vno della bracia della bilancia, ò dentro, ò &longs;uori, ò in qual &longs;i voglia modo trouere <lb/>mo le co&longs;e i&longs;te&longs;&longs;e.<emph.end type="italics"/></s> </p> <figure id="fig51"></figure> <p type="main"> <s>In que&longs;to luogo egli conuiene auertire, il che potcua&longs;i anco fare di &longs;opra à carte cin <lb/>que pre&longs;&longs;o la fine della &longs;econda faccia oue è &longs;critto. </s> <s>oltre à ciò po&longs;siamo con&longs;ide­<lb/>rare le co&longs;e che &longs;eguono in tutto al modo i&longs;te&longs;&longs;o. </s> <s>Che que&longs;to autore è &longs;tato il <lb/>primo à con&longs;iderare e&longs;qui&longs;itamente la bilancia, & intenderla dalla natura, & dal <lb/>vero e&longs;&longs;er &longs;uo; pero che egli il primiero di tutti ha manife&longs;tato chia ramente il mo <lb/>do del trattarla, & in&longs;egnarla, con proporre tre centri da e&longs;&longs;ere con&longs;iderati in que <lb/>&longs;ta &longs;peculatione; l'uno è il centro del mondo, l'altro il centro della bilancia, & il <lb/>terzo il centro della grauezza della bilancia, che in e&longs;&longs;a era vn na&longs;co&longs;to &longs;ecreto di <lb/>natura. </s> <s>Senza que&longs;ti tre centri, chiara co&longs;a è, che non &longs;i puote ve nire in cono&longs;ci­<lb/>mento per&longs;etto, ne dimo&longs;trare gli effetti varij della bilancia, i quali na&longs;cono dalla <lb/>diuer&longs;ità del collo care il centro della bilancia in tre modi, cioè quando il <lb/>centro della bilancia &longs;ta &longs;opra il centro della grauezza di e&longs;&longs;a, ouero quando è <lb/>di &longs;otto, o pure allhorche il centro della bilancia è nell'i&longs;te&longs;&longs;o centro della gra­<lb/>uezza di lei; &longs;i come l'autore in&longs;egna nella tre precedenti dimo&longs;trationi, cioè <lb/>nella <expan abbr="&longs;ecõda">&longs;econda</expan>, nella terza, & nella quarta propo&longs;itione: peroche nella &longs;econda mo­<lb/>&longs;tra quando la bilancia torna &longs;empre egualmente di&longs;tante dall'orizonte; nella ter­<lb/>za quando non &longs;olo non ritorna, ma &longs;i moue al contrario; nella quarta, che <lb/>e&longs;&longs;endo la bilancia &longs;o&longs;tenuta nel &longs;uo centro dalla grauezza &longs;ta ferma douunque el <lb/>la &longs;i troua, il quale effetto in particolare non è piu &longs;tato tocco, ne veduto, ne man <lb/>co da niuno manife&longs;tato, fuor che dall'autore: anzi fin hora tenuto fal&longs;o, & impo&longs; <lb/>&longs;ibile da tutti gli prede ce&longs;&longs;ori no&longs;tri; i quali con molte ragioni &longs;i &longs;ono sforzati di <lb/>prouare non &longs;olamente il contrario, ma hanno etiandio affermato per certo, che <lb/>la &longs;períenza mo&longs;tra la bilancia non dimorare gia mai ferma &longs;e non quando ella è <lb/>egualmente di&longs;tante dall'oritonte. </s> <s>Laqual co&longs;a in tutto è contraria alla ragione <lb/>prima, per e&longs;&longs;ere la dimo&longs;tratione della &longs;udetta quarta propo&longs;itione tanto chiara, <lb/>facile, & vera, che non sò, come &longs;e le po&longs;&longs;a in modo alcuno contradire: & poial­<lb/>l'e&longs;perienza concio&longs;ia che l'autore habbia fatto &longs;ottili&longs;simamente lauorare bilan­<lb/>cie giu&longs;te a po&longs;ta per chiarire que&longs;ta verità, vna delle quali hò io veduto in mano <lb/>dell'Illu&longs;tre Signor Gio. </s> <s>Vicenzo Pinello, mandatagli dall'i&longs;te&longs;&longs;o autore, la quale <lb/>per e&longs;&longs;ere &longs;o&longs;tenuta nel centro della &longs;u a grauezza, mo&longs;&longs;a douunque &longs;i vuole, & poi <lb/>la&longs;ciata, &longs;tà ferma in ogni &longs;ito doue ella vien la&longs;ciata. </s> <s>Ben è egli vero, che non bi <lb/>&longs;ogna, nel fare cote&longs;ta e&longs;perienza, correr co&longs;i a furia, per e&longs;&longs;ere co&longs;a oltra modo <lb/>difficile, come dice l'áutore di &longs;opra, il fare vna bilancia, la quale &longs;ia nel mezo del <lb/>le &longs;ue braccia &longs;o&longs;tenuta à punto, & nel centro proprio della &longs;ua grauezza. </s> <s>Per la <lb/>qual co&longs;a egli è da por <expan abbr="m&etilde;te">mente</expan>, che qual'hora alcuno &longs;i mette&longs;&longs;e à far cotale e&longs;perien <lb/>ra, & non gli riu&longs;ci&longs;&longs;e, non perciò &longs;i deue &longs;gomentare, anzi dica pur fermamente <lb/>di non hauer bene operato, & vn'altra volta ritorni à farne la &longs;perienza, fin chela <lb/>bilancia &longs;ia giu&longs;ta, & eguale, & venga &longs;o&longs;tenuta à punto nel centro della grauez­<lb/>za &longs;ua. </s> <s>Et benche da altri &longs;iano &longs;tate to cche le altre due predette &longs;peculationi, cioè <lb/>quando la bilancia ritorna &longs;empre egualmente di&longs;tante dall'orizonte, & quando <lb/>&longs;i moue al contrario di que&longs;to &longs;ito, tuttauia non &longs;i è piu inte&longs;a que&longs;ta verità gia <lb/>mai apertamente, &longs;e non dall'autore no&longs;tro; peroche gli altri non hanno co'l &longs;en­<lb/>no penetrato in ciò tanto auanti, che habbiano &longs;aputo con di&longs;tintione con&longs;idera <lb/>re il centro della bila ncia in tre modi, come hò narrato. </s> <s>Che &longs;e hanno pur diui&longs;a <lb/>to qualche co&longs;a d'intorno à que&longs;to, l'hanno fatto confu&longs;i&longs;simamente, & con ma <lb/>le dimo&longs;trationi, dalle quali non &longs;i puote cauare ferma <expan abbr="cõchiufione">conchiufione</expan>, & chiara. </s> <s>Que <lb/><gap/> predece&longs;&longs;ori no&longs;tri han&longs;i da intendere i moderni &longs;crittori di cotal materia alle­<lb/>gati in diuer&longs;i luoghi dall'autore, fra quali Giordano, che &longs;cri&longs;&longs;e de'pe&longs;i fù riputa- <pb pagenum="29"/>to a&longs;&longs;ai, & &longs;in qui è &longs;tato &longs;eguito molto nella &longs;ua dottrina. </s> <s>Hor l'autore no&longs;tro hà <lb/>procurato con ogni &longs;tudi o di caminare per la via de' buoni Greci antichi, <lb/>mae&longs;tri delle &longs;cienze, & in particolare di Archimede Sira cu&longs;ano prencipe delle ma <lb/>thematiche famo &longs;i&longs;simo, & di Pappo Ale&longs;&longs;andrino, come egli dice, leggendogli <lb/>nella &longs;ua propria fauella, non tradotti; peroche il piu delle volte &longs;ono co&longs;i mal <lb/>trattati, che à gran pen a'&longs;i puote trarre da loro frutto veruno. </s> <s>& affine che que&longs;ta <lb/>noua opinion &longs;ua, dimo&longs;trata à pien o nella predetta quarta propo&longs;itione, re&longs;ti to­<lb/>talmente chiara, non &longs;i è gia <expan abbr="cont&etilde;tato">contentato</expan> egli d'hauerla dimo&longs;trata con viue ragioni, <lb/>& certe &longs;olamente, ma come buon filo&longs;ofo, procedente con via di reale dottrina, <lb/>& di fon data &longs;cienza, (imitando Ari&longs;totele, ilqual ne' principii de &longs;uoi libri, inue­<lb/>&longs;tigando dottrina migliore, hà datto contra la opinione de gli antichi, &longs;oluendo <lb/>le ragioni addotte da loro:) hà ben voluto, e&longs;&longs;endo la verità vna &longs;ola, proporre le <lb/>opinioni de'&longs;uoi predece&longs;&longs;ori, & e&longs;aminare le loro ragioni, lequali &longs;embrano pro <lb/>uar il contrario, & &longs;oluerle, la loro fallenza <expan abbr="dimo&longs;trãdo">dimo&longs;trando</expan> co'l pre&longs;ente di&longs;cor&longs;o, che <lb/>incomincia, come è detto à carte cinque nella faccia &longs;econda, & qui fini&longs;ce ilqua <lb/>le di&longs;cor&longs;o &longs;eruirà in que&longs;ta materia, &longs;econdo che &longs;i &longs;uole dire per la opinione de <lb/>gli antichi. </s> <s>Et percio che egli contiene co&longs;e di alti&longs;sima &longs;peculatione, ma&longs;simamen­<lb/>te d'intorno al con&longs;iderare doue &longs;ia piu graue vn pe&longs;o &longs;olo po&longs;to in vno braccio <lb/>della bilancia, bi&longs;ogna in ogni modo, per bene intendere, leggerlo, & i&longs;tudiarlo <lb/>con accurati&longs;sima diligenza. </s> <s>Ma per certo l'autore è &longs;tato non &longs;olo il primo à tro <lb/>uare que&longs;ta verità, ma il primo etiandio a dim o&longs;trare in qual maniera &longs;ia me&longs;tieri <lb/>con&longs;iderare, & &longs;peculare interamente la pre&longs;ente materia tutta. </s> <s>Con laquale &longs;p ecu­<lb/>latione proua di nouo, & confermai varij effetti, & accidenti della bilancia già di <lb/>mo&longs;trati nelle pro&longs;sime tre propo&longs;itioni; mo&longs;trando ancora, come &longs;in qui cote&longs;te <lb/>co&longs;e &longs;iano da gli altri &longs;tate malamente con&longs;iderate, & con principij fal&longs;i. </s> <s>Anzi di <lb/>piu per confermatione della verità &longs;oggiunge, che que&longs;ti tali non hanno &longs;aputo fa <lb/>re le loro demo&longs;trationi; poi che co'l proprio modo di &longs;peculare v&longs;ato da loro, <lb/>& con le loro mede&longs;ime ragioni proua la &longs;ua intentione, & &longs;entenza e&longs;&longs;ere veri&longs;si <lb/>ma, appoggiando &longs;i alla dottrina di Ari&longs;totele &longs;empre, & facendo toccar con ma­<lb/>no, che egli con e&longs;&longs;o lui è d'accordo nelle que&longs;tioni mechaniche. </s> <s>In trattando <lb/>que&longs;ta materia moue l'autore alcuni dubbi molto belli, & curio&longs;i, & poi chiara­<lb/>ment e gli &longs;olue. </s> <s>In vltimo, accioche non manca&longs;&longs;e nulla al compiuto cono&longs;cimen <lb/>to di que&longs;to &longs;oggetto, egli hà trattato delle bilancie, che hanno le braccia di&longs;ugua <lb/>li, & di quelle che hanno le dette braccia piegate, & torte. </s> <s>In &longs;omma &longs;i può ben <lb/>affermare, che in cote&longs;to di&longs;cor&longs;o &longs;iano compre&longs;e tutte quelle co&longs;e, che po&longs;&longs;ono e&longs; <lb/>&longs;ere diui&longs;ate d'intorno à materia tale. </s> <s>Le quali &longs;ono di belli&longs;s ima & &longs;ottili&longs;sima &longs;pe <lb/>culatione, & à chiunque &longs;i diletta, & attende à que&longs;ti no bili &longs;tudi nece&longs;&longs;arij &longs;sime, <lb/>& da e&longs;&longs;ere, come hò ricordato piu d'una volta, con molta attentione vedute, & <lb/>con&longs;iderate. </s> </p> <p type="main"> <s>Doue &longs;i legge que&longs;to vocabolo latino Equilibrio, intenda&longs;i per eguale contrape&longs;o, <lb/>cio è che pe&longs;a tanto da vna banda, quanto dallaltra in pari lance, ò libra, ò bilancia <lb/>che &longs;i dica. </s> </p> <p type="main"> <s><emph type="italics"/>Librar congiu&longs;te lance.<emph.end type="italics"/></s> </p> <p type="main"> <s>Di&longs;&longs;e il Petrarcha. </s> </p> <pb pagenum="38"/> <p type="main"> <s><emph type="italics"/>FC &longs;ono eguali, &longs;imilmente i pe&longs;i FC pe&longs;eranno egualmente, ma i pe&longs;i FGC ap­<lb/>piccati nella leua EBA, il cui &longs;o&longs;tegno è in B non pe&longs;eranno egualmente; ma in­<lb/>chineranno in giu&longs;o dalla parte di A. </s> <s>Ponga&longs;i dunque in D tanta forza, che i <lb/>pe&longs;i FGC pe&longs;ino egualmente; &longs;arà la po&longs;&longs;anza in D eguale al pe&longs;o G; peroche<emph.end type="italics"/><lb/> <arrow.to.target n="fig52"></arrow.to.target><lb/><emph type="italics"/>i pe&longs;i FG pe&longs;ano egualmente, & la po&longs;&longs;anza in D niente altro deue fare, che <lb/>&longs;o&longs;tenere il pe&longs;o G che non di&longs;cenda. </s> <s>& percioche i pe&longs;i FGC, & la po&longs;&longs;anza <lb/>in D pe&longs;ano egualmente, leuati via dunque i pe&longs;i FG, i quali pe&longs;ano egualmente, <lb/>i re&longs;tanti pe&longs;eranno egualmente, cioè la po&longs;&longs;anza in D co'l pe&longs;o C, cioè la po&longs;&longs;an <lb/>za in D &longs;o&longs;terrà il pe&longs;o C, talche la leua AB stia come prima. </s> <s>& per e&longs;&longs;ere la <lb/>po&longs;&longs;anza in D eguale al pe&longs;o G, & il pe&longs;o C eguale al pe&longs;o, hauerà la po&longs;&longs;an <lb/>zaposta in D la proportione mede&longs;ima al pe&longs;o C, che EB, cioè AB à BD. <lb/>che bi&longs;ognaua mostrare.<emph.end type="italics"/></s> </p> <figure id="fig52"></figure> <p type="head"> <s>COROLLARIO I.</s> </p> <p type="main"> <s>Da que&longs;to è chiaro ancora, come prima, che &longs;e &longs;arà po&longs;to il pe­<lb/>&longs;o più vicino al &longs;o&longs;tegno B, come in H, il pe&longs;o douer&longs;i &longs;o­<lb/>&longs;tenere da forza minore. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche HB ha proportione minore à BD, che AB à BD. & quanto più<emph.end type="italics"/> <arrow.to.target n="note136"></arrow.to.target><lb/><emph type="italics"/>da vicino &longs;arà al &longs;o&longs;tegno, &longs;empre anco minore forza vi &longs;i ricercherà.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note136"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO II.</s> </p> <p type="main"> <s>Egli è parimente manife&longs;to, che la po&longs;&longs;anza in D è &longs;empre <lb/>maggiore del pe&longs;o C. </s> </p> <p type="main"> <s><emph type="italics"/>Perche &longs;e tra AB &longs;i piglia qual &longs;i voglia punto, come D, &longs;empre AB &longs;arà mag <lb/>giore di BD.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>El è da auertire, che que&longs;te dimo&longs;trationi lequali habbiamo prodotte in mezo, &longs;i po&longs;&longs;o­<lb/>no à tutte que&longs;te co&longs;e commodamente adattare non &longs;olamente e&longs;&longs;endo le leue egual­<lb/>mente distanti dall'orizonte, ma anche inchinate le dette leue all'orizonte. </s> <s>ilche è <lb/>chiaro da quel che nella bilancia &longs;i è diui&longs;ato.<emph.end type="italics"/></s> </p> <pb/> <p type="head"> <s>PROPOSITIONE IIII.</s> </p> <p type="main"> <s>Se la po&longs;&longs;anza mouerà il pe&longs;o appiccato nella leua, &longs;arà lo &longs;patio <lb/>della po&longs;&longs;anza mo&longs;&longs;a allo &longs;patio del pe&longs;o mo&longs;&longs;o, come la di&longs;tan <lb/>za dal &longs;o&longs;tegno alla po&longs;&longs;anza, alla di&longs;tanza dall'i&longs;te&longs;&longs;o &longs;o&longs;tegno <lb/>fin allo appiccamento del pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB, il cui &longs;o&longs;tegno C, & &longs;ia il pe&longs;o D attaccato al punto B, & &longs;ia <lb/>la po&longs;&longs;anza in A mouente il pe&longs;o D conlaleua AB. </s> <s>Dico lo &longs;patio della po&longs;­<lb/>&longs;anza in A allo &longs;patio del pe&longs;o e&longs;&longs;ere co&longs;i come CA à CB. </s> <s>Moua&longs;i la leua <lb/>AB, & affine che il pe&longs;o D &longs;i moua in sù, bi&longs;ogna che B &longs;i moua in sù, & A in <lb/>giù. </s> <s>& percioche C è punto immobile; però mentre A, & B &longs;i mouono, de­<lb/>&longs;criueranno circonferenze di cerchi. </s> <s>Moua&longs;i dunque AB in EF; &longs;aranno AEBF <lb/>circonferenze di cerchi, i me­<lb/>zi diametri de' quali &longs;ono CA <lb/>CB. </s> <s>compi&longs;ca&longs;i tutta la cir­<lb/>conferenza AGE, & tut­<lb/>ta la BHF, & &longs;iano KH <lb/>i punti doue AB, & EF ta­<lb/>gliano il cerchio BHF. </s> <s>Hor <lb/>percioche l'angolo BCF è<emph.end type="italics"/><lb/> <arrow.to.target n="note137"></arrow.to.target> <emph type="italics"/>eguale all'ango<gap/> HCK, &longs;a­<lb/>rà la circonferenza KH egua<emph.end type="italics"/><lb/> <arrow.to.target n="note138"></arrow.to.target> <emph type="italics"/>le alla circonferenza BF, & <lb/>concio&longs;ia, che le circonferen­<lb/>ze AEKH &longs;iano &longs;otto l'i­<lb/>&longs;te&longs;&longs;o angolo ACE, & la <lb/>circonferenza AE à tutta <lb/>la circonferenza AGE &longs;ia <lb/>come l'angolo ACE à quat <lb/>tro retti, & come l'i&longs;te&longs;&longs;o an­<lb/>golo HCK à quattro retti, <lb/>co&longs;i anche è la circonferenza <lb/>HK à tutta la circonferentia <lb/>HBK, &longs;arà la circonferentia<emph.end type="italics"/><lb/> <arrow.to.target n="fig53"></arrow.to.target><lb/><emph type="italics"/>AE à tutta la circonferentia AGE, come la circonferentia KH à tutta la<emph.end type="italics"/><lb/> <arrow.to.target n="note139"></arrow.to.target> <emph type="italics"/>KFH. & permutando come la circonferentia AE alla circonferenza KH, cioè <lb/>BF, co&longs;i tutta la circonferenza AGE à tutta la circonferenza BHF; ma tut-<emph.end type="italics"/><lb/> <arrow.to.target n="note140"></arrow.to.target> <emph type="italics"/>ta la circonferenza AGE co&longs;i &longs;i ha à tutta la BHF, come il diametro del cer­<lb/>chio AEG al diametro del cerchio BHF. </s> <s>Come dunque la circonferenza AE<emph.end type="italics"/> <pb pagenum="39"/><emph type="italics"/>ver&longs;o la circonferenza BF, co&longs;i è il diametro del cerchio AGE al diametro del<emph.end type="italics"/> <arrow.to.target n="note141"></arrow.to.target><lb/><emph type="italics"/>cerchio BHF: ma come il diametro al diametro, co&longs;i è anche il mezo diametro al <lb/>mezo diametro, cioè CA à CB. </s> <s>Per laqual co&longs;a come la circonferenza AE <lb/>alla circonferenza BF, co&longs;i CA à CB: ma la circonferenza AE è lo <lb/>&longs;patio della po&longs;&longs;anza mo&longs;&longs;a, & la circonferenza BF è eguale allo &longs;patio di D pe­<lb/>&longs;o mo&longs;&longs;o, peroche lo &longs;patio del mouimento del pe&longs;o D &longs;empre è eguale allo &longs;patio <lb/>del mouimento del punto B, per e&longs;&longs;ere attaccato in B. </s> <s>Lo &longs;patio dunque della po&longs; <lb/>&longs;anza mo&longs;&longs;a allo &longs;patio del pe&longs;o mo&longs;&longs;o è come CA à CB; cioè come la di&longs;tan­<lb/>za dal &longs;o&longs;tegno alla po&longs;&longs;anza, alla distanza dall'i&longs;te&longs;&longs;o all'appiccamento del pe&longs;o. </s> <s><lb/>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note137"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note138"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 26. <emph type="italics"/>del terzo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note139"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>del<emph.end type="italics"/> 15.</s> </p> <p type="margin"> <s><margin.target id="note140"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 23. <emph type="italics"/>del<emph.end type="italics"/> 8. <emph type="italics"/>di Pap po.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note141"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure id="fig53"></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;ia la leua AB, il cui &longs;o&longs;tegno B, & la po&longs;&longs;anza mouente in A, & il pe&longs;o <lb/>in C. </s> <s>Dico lo &longs;patio della po&longs;&longs;anza mo&longs;&longs;a allo &longs;patio del pe&longs;o tra&longs;portato co&longs;i e&longs;­<lb/>&longs;ere, come BA à BC. </s> <s><lb/>Moua&longs;i la leua, & accioche <lb/>il pe&longs;o &longs;ia alzato in sù, egli <lb/>è nece&longs;&longs;ario, che anche i pun <lb/>ti CA &longs;imouano in sù. </s> <s><lb/>Moua&longs;i dunque A in sù <lb/>fin'in D; & &longs;ia il mouimen <lb/>to della leua BD. </s> <s>mo&longs;tre­<lb/>remo nel modo i&longs;te&longs;&longs;o, come <lb/>prima è detto, che i punti <lb/>CA de&longs;criuono circonferen <lb/>ze di cerchi, i cui mezi dia­<lb/>metri &longs;ono BA BC. & di­<lb/>mo&longs;treremo &longs;imilmente co&longs;i <lb/>e&longs;&longs;ere AD à CE, come il <lb/>mezo diametro AB al me­<lb/>zo diametro BC.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et per la ragione i&longs;te&longs;&longs;a, &longs;e la <lb/>po&longs;&longs;anza fo&longs;&longs;e in C, & il <lb/>pe&longs;o in A &longs;i prouerà co&longs;i <lb/>e&longs;&longs;ere CE ver&longs;o AD, co-<emph.end type="italics"/><lb/> <arrow.to.target n="fig54"></arrow.to.target><lb/><emph type="italics"/>me BC à BA, cioè la di&longs;tanza dal &longs;o&longs;tegno alla po&longs;&longs;anza; alla di&longs;tanza dal­<lb/>l'iste&longs;&longs;o allo attaccamento del pe&longs;o. </s> <s>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig54"></figure> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Da que&longs;te co&longs;e è manife&longs;to, che maggiore proportione ha lo &longs;pa <lb/>tio della po&longs;&longs;anza, che moue allo &longs;patio del pe&longs;o mo&longs;&longs;o, che il <lb/>pe&longs;o alla mede&longs;ima po&longs;&longs;anza. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche lo &longs;patio della po&longs;&longs;anza allo &longs;patio del pe&longs;o ha la mede&longs;ima proportione, <lb/>che il pe&longs;o alla po&longs;&longs;anza, che &longs;o&longs;liene il detto pe&longs;o. </s> <s>Ma la po&longs;&longs;anza, che &longs;ostie­<lb/>ne è minore della po&longs;&longs;anza che moue, però il pe&longs;o haurà proportione minore alla <lb/>po&longs;&longs;anza che lo moue, che alla po&longs;&longs;anza, che lo &longs;ostiene. </s> <s>Lo &longs;patio dunque della<emph.end type="italics"/><lb/> <arrow.to.target n="note142"></arrow.to.target> <emph type="italics"/>po&longs;&longs;anza che moue allo &longs;patio del pe&longs;o haur à proportione maggiore, che il pe&longs;o al­<lb/>l'i&longs;le&longs;&longs;a po&longs;&longs;anza.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note142"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE V.</s> </p> <p type="main"> <s>La po&longs;&longs;anza che in qual &longs;i voglia modo &longs;o&longs;tenga il pe&longs;o con la le­<lb/>ua hauerà la proportione mede&longs;ima ad e&longs;&longs;o pe&longs;o, che la di&longs;tan <lb/>za frapo&longs;ta dal &longs;o&longs;tegno al punto, doue dal centro della gra­<lb/>uezza del pe&longs;o tirata vna linea à piombo all'orizonte tagli la <lb/>leua, alla di&longs;tanza che è fra il &longs;o&longs;tegno, & la po&longs;&longs;anza. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB egualmente di&longs;tante dall'orizonte, col &longs;uo &longs;o&longs;tegno N. </s> <s>&longs;ia dopo il pe <lb/>&longs;o AC, il cui centro della grauezza &longs;ia D, ilquale &longs;ia prima &longs;otto la leua: ma <lb/>il pe&longs;o &longs;ia appiccato à i punti AO. & dal punto D &longs;ia tirata la linea DE à <lb/>piomho dell' orizonte, & di AB. </s> <s>Che &longs;e vi &longs;aranno altre leue ancora AF AG, <lb/>i cui &longs;o­<lb/>stegni, <lb/>&longs;iano H <lb/>K, & il <lb/>pe&longs;o A <lb/>C &longs;ia ap <lb/>piccato <lb/>nella le­<lb/>ua AG <lb/>ne i pun <lb/>ti AQ, <lb/>& nella <lb/>leua A <lb/>F ne' <expan abbr="pũ">pum</expan> <lb/>ti AP: <lb/>& la li­<lb/>nea DE <lb/>allunga-<emph.end type="italics"/><lb/> <arrow.to.target n="fig55"></arrow.to.target><lb/><emph type="italics"/>ta tagli AF in L, & AC in M. </s> <s>Dico che la po&longs;&longs;anza in F &longs;o&longs;tenente il pe&longs;o AC <lb/>ha quella proportione ad e&longs;&longs;o pe&longs;o, che ha KL à KF; & la po&longs;&longs;anza in D ha quella <lb/>proportione al pe&longs;o, che ha NE ad NB; & la po&longs;&longs;anza in G al pe&longs;o quella, che ha <lb/>HM ad HG. </s> <s>Hor percioche DL &longs;tà à piombo dell' orizonte, il pe&longs;o AC venga ap-<emph.end type="italics"/> <pb pagenum="40"/><emph type="italics"/>piccato doue &longs;i voglia nella linea DL, rimarrà nel modo i&longs;te&longs;&longs;o che &longs;i troua. </s> <s>Per la­<lb/>qual co&longs;a &longs;e nella leua AB &longs;i &longs;cioglieranno gli appiccamenti, che &longs;ono ad AO, il <lb/>pe&longs;o AC appiccato in E rimarr à nell'i&longs;te&longs;&longs;o modo, come hora rimane, cioè leuato via <lb/>il punto A, & la linea QO, nell'i&longs;te&longs;&longs;o modo il pe&longs;o appiccato in E rimarrà, come <lb/>era &longs;o&longs;tenuto da punti i&longs;te&longs;&longs;i AO, come &longs;i proua per lo commentario di Federico <lb/>Commandino nella &longs;esta propo&longs;itione di Archimede della quadratura della parabo <lb/>la, & dalla prima di que&longs;to della bilancia. </s> <s>Co&longs;i percio che il pe&longs;o AC ha &longs;empre la <lb/>i&longs;te&longs;&longs;a di&longs;po&longs;itione ver&longs;o la bilancia, &longs;ia pur in AO &longs;ostentato, ouero pendente dal <lb/>punto E; la po&longs;&longs;anza mede&longs;ima in B &longs;o&longs;tenterà il pe&longs;o i&longs;te&longs;&longs;o AC pendente, ouero <lb/>da E, ouero da AO. ma la po&longs;&longs;anza in B &longs;o&longs;tenente il pe&longs;o AC appiccato in E co&longs;i <lb/>&longs;i hà ad e&longs;&longs;o pe&longs;o, come NE ad NB; La po&longs;&longs;anza dunque in B &longs;o&longs;tenente il pe&longs;o <lb/>AC da punti AO pendente &longs;arà co&longs;i ad e&longs;&longs;o pe&longs;o, come NE ad NB. </s> <s>Non altra-<emph.end type="italics"/> <arrow.to.target n="note143"></arrow.to.target><lb/><emph type="italics"/>mente &longs;i mo&longs;trerà, che il pe&longs;o AC pendente dal punto L rimane, come &longs;e &longs;o&longs;&longs;e &longs;oste <lb/>nuto da punti AP; & la po&longs;&longs;anza in F ad e&longs;&longs;o pe&longs;o e&longs;&longs;ere co&longs;i come<emph.end type="italics"/> KL <emph type="italics"/>à KF. </s> <s>Ma <lb/>nella leua AG il pe&longs;o AC appiccato in M co&longs;i rimanere, come egli è &longs;o&longs;tenuto da <lb/>punti <expan abbr="Aq;">Aque</expan> & la po&longs;&longs;anza di G co&longs;i e&longs;&longs;ere al pe&longs;o AC, come HM ad HG, cioè co­<lb/>me la di&longs;tanza dal &longs;o&longs;tegno al punto, doue la linea tirata à piombo dell' orizonte <lb/>dal centro della grauezza del pe&longs;o taglia la leua, alla di&longs;tanza dal &longs;o&longs;tegno alla po&longs;­<lb/>&longs;anza. </s> <s>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note143"></margin.target><emph type="italics"/>Per la prima di questo.<emph.end type="italics"/></s> </p> <figure id="fig55"></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e FBG &longs;o&longs;&longs;ero i &longs;o&longs;tegni delle leue, & le po&longs;&longs;anze fo&longs;&longs;ero in KNH &longs;o&longs;tenenti il pe <lb/>&longs;o, con &longs;imile modo &longs;i mo&longs;trerà la po&longs;&longs;anza in H, co&longs;i e&longs;&longs;ere al pe&longs;o, come GM à GH, <lb/>et la <expan abbr="po&longs;sāzaī">po&longs;sanzaim</expan> N al pe&longs;o, come BE à BN, et la <expan abbr="po&longs;sāza">po&longs;sanza</expan> <expan abbr="ī">im</expan> K al pe&longs;o come FL ad FK.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et &longs;e le leue AB AF AG haue&longs;&longs;ero i &longs;o&longs;legni in A, & il pe&longs;o fo&longs;&longs;e NO; poi dal <lb/>centro D del <lb/>la &longs;ua gra­<lb/>uezza fo&longs;&longs;e <lb/>tirata la li­<lb/>nea DME <lb/>L à piombo <lb/>di AB, & <lb/>dell' orizon <lb/>te, & fo&longs;&longs;e­<lb/>ro le po&longs;&longs;an <lb/>ze in FB <lb/>G; <expan abbr="&longs;imilm&etilde;">&longs;imilmen</expan> <lb/>re mo&longs;tre­<lb/>ra&longs;&longs;i la po&longs; <lb/>&longs;anza di G <lb/>&longs;o&longs;tenente <lb/>il pe&longs;o N<emph.end type="italics"/><lb/> <arrow.to.target n="fig56"></arrow.to.target><lb/><emph type="italics"/>O co&longs;i e&longs;&longs;ere ad e&longs;&longs;o pe&longs;o, come AM ad AG, & la po&longs;&longs;anza in B come AE ad <lb/>AB; & la po&longs;&longs;anza in F come AL ad AF.<emph.end type="italics"/></s> </p> <figure id="fig56"></figure> <p type="main"> <s><emph type="italics"/>Sia dapoi la leua AB egualmente di&longs;tante dall'orizonte, il cui &longs;o&longs;tegno &longs;ia D, & &longs;ia <lb/>BE il pe&longs;o, il cui centro della grauezza &longs;ia F &longs;opra la leua; & dal punto F riri&longs;i la <lb/>linea FH à piombo, & dell' orizonte, & di e&longs;&longs;a AB; & &longs;ia &longs;o&longs;tenuto il pe&longs;o dal <lb/>punto B, & da <expan abbr="Pq.">Pque</expan> &longs;iano po&longs;cia altre leue BLBM, i cui &longs;o&longs;tegni &longs;iano NO; <lb/>& la linea FH allungatatagli BM in K, & BL in G; & venga &longs;o&longs;tenuto il pe&longs;o<emph.end type="italics"/><lb/> <arrow.to.target n="fig57"></arrow.to.target><lb/><emph type="italics"/>nella leua BL ne'punti BP; & nella leua BM dal punto B, & PR. </s> <s>Dico, che <lb/>la po&longs;&longs;anza in L &longs;o&longs;tenente il pe&longs;o BE nella leua BL ha quella proportione ad <lb/>e&longs;&longs;o pe&longs;o, che NG ad NL; & la po&longs;&longs;anza in A al pe&longs;o ha quella proportio­<lb/>ne, che DH à DA; & la po&longs;&longs;anza di M al pe&longs;o ha quella proportione, che <lb/>OK ad OM. </s> <s>Hor percioche la linea KF tirata dal centro della grauezza F è<emph.end type="italics"/><lb/> <arrow.to.target n="note144"></arrow.to.target> <emph type="italics"/>à piombo dell' orizonte, &longs;ia pur &longs;ostenuto il pe&longs;o da qual &longs;i voglia punto della linea <lb/>KF, egli rimarrà, come hora &longs;i troua. </s> <s>Se dunque &longs;arà &longs;ostenuto in H, rimarrà co <lb/>me prima, cioè leuato via il punto B, & PQ, i quali &longs;o&longs;tengono il pe&longs;o, rimarrà <lb/>il pe&longs;o BE nel modo che da e&longs;&longs;i era &longs;o&longs;tenuto. </s> <s>Per la qual co&longs;a grauerà nella le­<lb/>ua AB in H, & haurà alla leua quella di&longs;po&longs;itione mede&longs;ima, che prima, & per­<lb/>ciò &longs;arà come &longs;e fo&longs;&longs;e appiccato in H. </s> <s>La mede&longs;ma po&longs;&longs;anza dunque &longs;o&longs;terrà il me <lb/>de&longs;imo pe&longs;o BE &longs;o&longs;tentato ouero in H, ouero in B & <expan abbr="q.">que</expan> Ma la po&longs;&longs;anza in A<emph.end type="italics"/><lb/> <arrow.to.target n="note145"></arrow.to.target> <emph type="italics"/>&longs;o&longs;tenente il pe&longs;o BE appiccato in H con la leua AB hal'i&longs;te&longs;&longs;a proportione ad e&longs;­<lb/>&longs;o pe&longs;o, che DH à DA; l'i&longs;te&longs;&longs;a po&longs;&longs;anza dunque in A &longs;o&longs;tenente il pe&longs;o BE ne' <lb/>punti BQ &longs;o&longs;tentato, &longs;arà ad e&longs;&longs;o pe&longs;o come DH à DA. </s> <s>Similmente &longs;i mo&longs;trer à <lb/>il pe&longs;o BE, &longs;e in G &longs;arà &longs;o&longs;tenuto, rimanere come egliera &longs;o&longs;tenuto da punti BP: <lb/>& nel punto K, come dapunti BR. </s> <s>Per la qual co&longs;a la po&longs;&longs;anza in L &longs;o&longs;tenente<emph.end type="italics"/> <pb pagenum="41"/><emph type="italics"/>il pe&longs;o BE ad e&longs;&longs;o pe&longs;o co&longs;i &longs;arà come NG ad NL. ma la po&longs;&longs;anza in M al pe <lb/>&longs;o, come OK ad OM; cioè come la di&longs;tanza dal &longs;o&longs;tegno al punto, doue dal cen <lb/>tro della grauezza del pe&longs;o la linea tirata à piombo dell' orizonte taglia la leua, alla <lb/>di&longs;tanza dal &longs;o&longs;tegno alla po&longs;&longs;anza. </s> <s>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note144"></margin.target><emph type="italics"/>Per la prima di questo della bilancia.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note145"></margin.target><emph type="italics"/>Per la prima di questo.<emph.end type="italics"/></s> </p> <figure id="fig57"></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e LAM fo&longs;&longs;ero i &longs;o&longs;tegni, & le po&longs;&longs;anze in NDO; &longs;imilmente mo&longs;trera&longs;&longs;i <lb/>la po&longs;&longs;anza in N co&longs;i e&longs;&longs;ere al pe&longs;o, come LG ad LN; & la po&longs;&longs;anza in D, <lb/>come AH ad AD, & la po&longs;&longs;anza in O come MK ad MO.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et &longs;e le leue BA BL BM haue&longs;&longs;ero i &longs;o&longs;tegni in B, & il pe&longs;o fo&longs;&longs;e NO &longs;opra <lb/>la leua, & dal centro F della grauezza fo&longs;&longs;e tirata la linea FD EG à piombo <lb/>di AB, & dell' orizonte; & fo&longs;&longs;ero le po&longs;&longs;anze in LAM, &longs;imilmente proue-<emph.end type="italics"/><lb/> <arrow.to.target n="fig58"></arrow.to.target><lb/><emph type="italics"/>ra&longs;&longs;i la po&longs;&longs;anza in L &longs;o&longs;tenente il pe&longs;o co&longs;i e&longs;&longs;ere ad e&longs;&longs;o pe&longs;o, come BD à BL; <lb/>& la po&longs;&longs;anza in A al pe&longs;o come BE à BA, & la po&longs;&longs;anza in M come BG <lb/>à BM.<emph.end type="italics"/></s> </p> <pb/> <figure id="fig58"></figure> <p type="main"> <s><emph type="italics"/>Sia vltimamente la leua. </s> <s>AB egualmente di&longs;tante dall'orizonte, il cui &longs;ostegno &longs;ia <lb/>C, & il pe&longs;o DE habb<gap/>a il centro della graueza F nella leua AB; & &longs;iano <lb/>alla fine altre leue GHKL, co i &longs;o&longs;tegni &longs;uoi MN; & il pe&longs;o nella leua GH <lb/>&longs;ia &longs;o&longs;tentato da i punti GO, & nella leua AB da punti AT, & nella leua <lb/>KL da punti KQ, & il centro F della grauezza &longs;ia parimente in amendue le le-<emph.end type="italics"/><lb/> <arrow.to.target n="fig59"></arrow.to.target><lb/><emph type="italics"/>ue GH<emph.end type="italics"/> K<emph type="italics"/>L, & &longs;iano le po&longs;&longs;anze in HBL. </s> <s>Dico la po&longs;&longs;anza in H co&longs;i e&longs;&longs;ere al <lb/>pe&longs;o, come N<emph.end type="italics"/>F <emph type="italics"/>ad NH; & la po&longs;&longs;anza in B alpe&longs;o, come C<emph.end type="italics"/>F <emph type="italics"/>à CB, & la po&longs; <lb/>&longs;anza in L alpe&longs;o, come M<emph.end type="italics"/>F <emph type="italics"/>ad ML. </s> <s>Hor percioche F è il centro della grauez­<lb/>za del pe&longs;o DE, &longs;e dunque in<emph.end type="italics"/> F <emph type="italics"/>&longs;arà &longs;o&longs;tenuto, &longs;tarà il pe&longs;o DE come prima, per <lb/>la diffinitione del centro della grauezza; & &longs;arà come &longs;e egli fo&longs;&longs;e appiccato in<emph.end type="italics"/> F; <lb/><emph type="italics"/>& &longs;tarà nella leua in quel modo i&longs;te&longs;&longs;o, &longs;o&longs;tenga&longs;i pure ò da punti AP, ouero dal <lb/>punto<emph.end type="italics"/> F. <emph type="italics"/>ilche parimente auerrà nelle leue GH KL, cioè che il pe&longs;o re&longs;terà nel mo <lb/>do i&longs;te&longs;&longs;o, &longs;o&longs;tenti&longs;i pur ò in<emph.end type="italics"/> F, <emph type="italics"/>ouero in GO ouero in <expan abbr="Kq.">Kque</expan> La mede&longs;ma po&longs;&longs;anza <lb/>dunque in B &longs;o&longs;tenterà il pe&longs;o i&longs;te&longs;&longs;o DE appiccato, ouero in<emph.end type="italics"/> F, <emph type="italics"/>ouero in AP: & <lb/>quando egli è appiccato in<emph.end type="italics"/> F, <emph type="italics"/>è ad e&longs;&longs;o pe&longs;o come CF à CB, dunque la po&longs;&longs;anza &longs;o­<lb/>&longs;tenente il pe&longs;o DE appiccato ad AP &longs;arà ad e&longs;&longs;o pe&longs;o come C<emph.end type="italics"/>F <emph type="italics"/>à CB. & nel mo <lb/>do i&longs;te&longs;&longs;o <expan abbr="lã">lam</expan> po&longs;&longs;anza in H &longs;arà al pe&longs;o appiccato in OG co&longs;i, come N<emph.end type="italics"/>F <emph type="italics"/>ad NH. & <lb/>la po&longs;&longs;anza in L &longs;arà al pe&longs;o appiccato in KQ, come M<emph.end type="italics"/>F <emph type="italics"/>ad ML. ilche anco bi&longs;o­<lb/>gnaua mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig59"></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e li &longs;o&longs;tegni fo&longs;&longs;ero HBL, & le po&longs;&longs;anze fo&longs;&longs;ero in NCM; &longs;imilmente prouera&longs;&longs;i <lb/>la po&longs;&longs;anza in N co&longs;i e&longs;&longs;ere al pe&longs;o, come HF ad HN & la po&longs;&longs;anza in C come <lb/>BF à BC; & la po&longs;&longs;anza in M come LF ad LM.<emph.end type="italics"/></s> </p> <pb pagenum="42"/> <p type="main"> <s><emph type="italics"/>Et &longs;e le leue BA BC BD haue&longs;&longs;ero i &longs;o&longs;tegni in B, & fo&longs;&longs;ero i pe&longs;i in EF GH <lb/>KL, di modo che i loro centri della grauezza MNO fo&longs;&longs;ero nelle leue, & le<emph.end type="italics"/><lb/> <arrow.to.target n="fig60"></arrow.to.target><lb/><emph type="italics"/>po&longs;&longs;anze fo&longs;&longs;ero in CAD. </s> <s>Similmente prouera&longs;&longs;i, che la po&longs;&longs;anza in C co&longs;i è <lb/>al pe&longs;o EF, come BM à BC, & la po&longs;&longs;anza in A al pe&longs;o GH, come <lb/>BN à BA, & la po&longs;&longs;anza in D al pe&longs;o KL, come BO à BD.<emph.end type="italics"/></s> </p> <figure id="fig60"></figure> <p type="head"> <s>PROPOSITIONE VI.</s> </p> <p type="main"> <s>Sia AB linea retta, ad angoli retti, dellaquale &longs;tia AD, la­<lb/>quale dalla parte di D &longs;ia allungata come &longs;i vuole fin'al C, <lb/>& &longs;ia congiunta la CB, laquale parimente allunghi&longs;i dalla <lb/>parte di B fin ad E. </s> <s>Dapoi &longs;iano dal punto B tirate altre <lb/>linee, come &longs;i vuole BF BG eguali ad AB tra AB BE; <lb/>& da i punti FG &longs;iano tirate le linee FH GK à piombo <lb/>delle &longs;udette, lequali &longs;i facciano eguali fra loro, & ad e&longs;&longs;a A <lb/>D come &longs;e BA AD fo&longs;&longs;ero mo&longs;&longs;e in BF FH, & in BG <lb/>GH; & congiungan&longs;i CH CK, lequali taglino le linee BF <lb/>BG ne'punti MN. </s> <s>Dico che BN è minore di BM, & <lb/>BM di e&longs;&longs;a BA. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Congiungan&longs;i BD BH B<emph.end type="italics"/>K, <emph type="italics"/>& percioche due linee DA AB &longs;ono eguali à due<emph.end type="italics"/><lb/> <arrow.to.target n="note146"></arrow.to.target> <emph type="italics"/>HF FB, & l'angolo DAB retto è anco eguale al retto HFB; &longs;aranno i <lb/>re&longs;tanti angoli eguali à i re&longs;tanti angoli, & HB eguale ad e&longs;&longs;a DB. </s> <s>Similmen­<lb/>te mo&longs;trera&longs;&longs;i il triangolo BKG e&longs;&longs;ere eguale al triangolo BHF. </s> <s>Per laqual co <lb/>&longs;a co'l centro B, & con l'in <lb/>teruallo di vna di e&longs;&longs;e de&longs;cri­<lb/>ua&longs;i il cerchio DH<emph.end type="italics"/> K<emph type="italics"/>E, il <lb/>quale tagli le linee CH CK <lb/>ne' punti OP; & congiun­<lb/>gan&longs;i OB PB. </s> <s>Percioche <lb/>dunque il punto K è più vi­<lb/>cino ad E, che H; &longs;arà la<emph.end type="italics"/><lb/> <arrow.to.target n="note147"></arrow.to.target> <emph type="italics"/>linea CK maggiore di CH, <lb/>& CP minore di CO: dun <lb/>que PK &longs;arà maggiore di <lb/>OH. </s> <s>Ma perche il triangolo <lb/>BKP di due lati eguali ha i <lb/>&longs;uoi lati BK BP eguali à i <lb/>lati BH BO del triangolo <lb/>BHO di due lati eguali, ma <lb/>ben la ba&longs;e KP maggiore <lb/>della ba&longs;e HO, &longs;arà l'ango­<lb/>lo KBP maggiore dell' an<emph.end type="italics"/><lb/> <arrow.to.target n="note148"></arrow.to.target> <emph type="italics"/>golo HBO. dunque i restan <lb/>ti angoli alla ba&longs;e, cioè KPB <lb/>PKB pre&longs;i in&longs;ieme, i quali <lb/>tra loro &longs;ono eguali, &longs;aranno <lb/>minori de i re&longs;tanti angol i al­<lb/>la ba&longs;e po&longs;ti, cioè OHB <lb/>HOB, iquali etiandio tra lo <lb/>ro &longs;ono eguali e&longs;&longs;endo che tut <lb/>ti gli angoli di cia&longs;cuno trian<emph.end type="italics"/><lb/> <arrow.to.target n="note149"></arrow.to.target> <emph type="italics"/>golo &longs;iano eguali à due angoli <lb/>retti. </s> <s>Per laqual co&longs;a anche <lb/>le metà di que&longs;ti, cioè NKB <lb/>&longs;arà minore di MHB. </s> <s>Et<emph.end type="italics"/><lb/> <arrow.to.target n="note150"></arrow.to.target> <emph type="italics"/>concio&longs;ia, che l'angolo BKG<emph.end type="italics"/><lb/> <arrow.to.target n="fig61"></arrow.to.target><lb/><emph type="italics"/>&longs;ia eguale all'angolo BHF, &longs;arà NKG maggiore di MHF. </s> <s>Se dunque nel <lb/>punto K &longs;i &longs;accia l'angolo GKQ eguale ad FHM &longs;i &longs;arà il triangolo GKQ <lb/>eguale al triangolo FHM; Imperoche due angoli in FH divno &longs;ono eguali à <lb/>due in GK d'vn'altro, & il lato FH è eguale al lato GK, &longs;arà GQ eguale <lb/>ad FM. </s> <s>Adunque GN &longs;arà maggiore di FM. & co&longs;i per e&longs;&longs;ere BG egua-<emph.end type="italics"/> <pb pagenum="43"/><emph type="italics"/>le à BF, &longs;arà BN minore di e&longs;&longs;a BM. ma che BM &longs;ia minore di e&longs;&longs;a BA <lb/>è manife&longs;to, percioche BM, è minore di e&longs;&longs;a BF, laquale è eguale à BA. che <lb/>bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note146"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note147"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del serzo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note148"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 25. <emph type="italics"/>del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note149"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note150"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 26. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure id="fig61"></figure> <p type="main"> <s><emph type="italics"/>Di più &longs;e tra BG BE &longs;i tiri à piacere vn'altra linea eguale à BG; & faccia&longs;i l'ope <lb/>ratione, come di &longs;opra è stato detto, prouera&longs;&longs;i &longs;imilmente la linea BR e&longs;&longs;er mi­<lb/>nore di BN. & quanto più da vicino &longs;arà ad e&longs;&longs;a BE, &longs;arà anche &longs;empre minore.<emph.end type="italics"/></s> </p> <p type="main"> <s>Che &longs;e i triangoli eguali BFH BGK fo&longs;&longs;ero di &longs;otto fra BC <lb/>BA collocati; & fo&longs;&longs;ero congiunte le linee HC KC, le­<lb/>quali taglia&longs;&longs;ero le linee BF BG allungate dalla parte di FG <lb/>ne' punti MN, &longs;arà <lb/>la BN maggiore del <lb/>la BM, & la BM di <lb/>e&longs;&longs;a BA. </s> </p> <p type="main"> <s><emph type="italics"/>Imperoche allunghi&longs;i CH CK <lb/>fin alla circonferenza in OP, <lb/>& congiungan&longs;i BO BP; <lb/>con &longs;imile modo mo&longs;trera&longs;&longs;i <lb/>la linea PK e&longs;&longs;ere maggiore <lb/>ai OH, & l'angolo PKB e&longs; <lb/>&longs;ere minore dell <expan abbr="ãgolo">angolo</expan> OH</s> <s>/>& percioche l'angolo BHF <lb/>è eguale dell' angolo BKG, &longs;a <lb/>rà tutto l'angolo PKG mi­<lb/>nore dell' angolo OHF. </s> <s>Per <lb/>laqual co&longs;a il re&longs;tante GKN <lb/>&longs;arà maggiore del re&longs;tante <lb/>FHM. </s> <s>Se <expan abbr="dũque">dunque</expan> fara&longs;&longs;i l'an <lb/>golo GKQ eguale ad FHM <lb/>la linea KQ taglier à in modo <lb/>la GN, che GQ diuenterà <lb/>eguale ad FM. </s> <s>Per laqual <lb/>co&longs;a maggiore &longs;arà GN, che <lb/>FM; allequali &longs;e &longs;ar anno ag <lb/>giunte le eguali BF BG, &longs;a­<lb/>rà BN maggiore di BM. & <lb/>per e&longs;&longs;ere BM maggiore di <lb/>FB, &longs;arà anco maggiore di <lb/>BA. </s> <s>&longs;imilmente prouera&longs;&longs;i <lb/>che <expan abbr="quãto">quanto</expan> più da vicino &longs;arà <lb/>BG à BC, la linea BN &longs;em <lb/>pre &longs;arà maggiore.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>PROPOSITIONE VII.</s> </p> <p type="main"> <s>Sia la linea retta AB, à cui &longs;tia à piombo AD, laquale allun­<lb/>ghi&longs;i dalla parte di D come pare &longs;in'à C, & congiunga&longs;i C <lb/>B, laquale etiandio &longs;i allunghi fin'ad E; & &longs;imilmente tra <lb/>AB BE &longs;iano, come pare, tirate BF BG eguali ad e&longs;&longs;a AB, <lb/>& da punti FG &longs;iano <lb/>tirate le linee FH GK <lb/>pur eguali ad e&longs;&longs;a AD, <lb/>& à piombo di BF BG, <lb/>come &longs;e BA AD fo&longs;­<lb/>&longs;ero mo&longs;&longs;e in BF FH <lb/>BG GK: & congiun­<lb/>gan&longs;i CH CK, lequali <lb/>taglino le linee allunga <lb/>te BF BG ne' punti <lb/>MN. </s> <s>Dico che BN è <lb/>maggiore di BM, & <lb/>BM di e&longs;&longs;a BA. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Congiungan&longs;i BD BH BK, & <lb/>co'l centro B, & conlo &longs;patio <lb/>BD de&longs;criua&longs;i it cerchio. </s> <s>&longs;imil­<lb/>mente come nella precedente, di­<lb/>mo&longs;treremo ipunti KHDOP <lb/>e&longs;&longs;ere nella circonferenza del cer <lb/>chio; & <expan abbr="itriāgoli">itriangoli</expan> ABD FBH <lb/>GBK e&longs;&longs;ere tra loro eguali, & <lb/>la linea PK e&longs;&longs;ere maggiore <lb/>della OH, & l'angolo PKB <lb/>e&longs;&longs;ere minore dell'angolo OHB. </s> <s><lb/>Percioche <expan abbr="dũque">dunque</expan> l'angolo BHF <lb/>è eguale all'angolo BKG, &longs;arà <lb/>tutto l'angolo PKG minore <lb/>dell'angolo OHF. </s> <s>Per laqual <lb/>co&longs;a il re&longs;tante G<emph.end type="italics"/>K<emph type="italics"/>N &longs;arà <lb/>maggiore del re&longs;tante FHM. </s> <s><lb/>Se <expan abbr="dũque">dunque</expan> &longs;i &longs;arà l'angolo G<emph.end type="italics"/>K<emph type="italics"/>Q<emph.end type="italics"/> <pb pagenum="44"/><emph type="italics"/>eguale ad e&longs;&longs;o FHM, &longs;arà il triangolo GKQ eguale al triangolo FHM, & <lb/>illato GQ al lato FM eguale; &longs;arà dunque maggiore GN di e&longs;&longs;a FM; & <lb/>perciò BN maggiore &longs;arà di BM. & BM &longs;arà maggiore di BA; impe­<lb/>roche BM èmaggiore di e&longs;&longs;ai BE. </s> <s>Che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et nel modo i&longs;te&longs;&longs;o in tutto, quanto più da pre&longs;&longs;o &longs;arà BG ad e&longs;&longs;a BE, &longs;empre la li­<lb/>nea BN &longs;i dimo&longs;trerà e&longs;&longs;er maggiore.<emph.end type="italics"/></s> </p> <p type="main"> <s>Che &longs;e &longs;aranno po&longs;ti di <lb/>&longs;otto i triangoli BF <lb/>HB GK tra AB <lb/>BC, & &longs;iano tiratele <lb/>linee CHO GKP, <lb/>lequali taglino le li­<lb/>nee BF BG ne' pun <lb/>ti MN: &longs;arà la linea <lb/>BN minore di e&longs;&longs;a <lb/>BM, & BM di e&longs;sa <lb/>BA. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Congiungan&longs;i BO BP. </s> <s>&longs;imilmen <lb/>te prouera&longs;&longs;i, che l'angolo P <lb/>KB è minore dell' angolo OH <lb/>B. </s> <s>Hor percioche l'angolo F <lb/>HB è eguale all'angolo GKB; <lb/>&longs;arà l'angolo GKN maggio­<lb/>re dell'angolo FHM: per la <lb/>qual co&longs;a la linea GN &longs;arà <lb/>maggiore di e&longs;&longs;a FM. & per­<lb/>ciò la linea BN &longs;arà minore <lb/>della linea BM. & concio­<lb/>&longs;ia che maggiore &longs;ia BF di <lb/>BM; &longs;arà BM minore di <lb/>BA. </s> <s>& con &longs;imile modo <lb/>proueraßi, che quanto più B <lb/>G &longs;arà dapre&longs;&longs;o ad e&longs;&longs;a BC, <lb/>la linea BN &longs;empre &longs;arà <lb/>minore.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE VIII.</s> </p> <p type="main"> <s>La po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o che habbia il centro della grauez­<lb/>za fopra la leuà egualmente di&longs;tante dall'orizonte, quanto <lb/>più il pe&longs;o &longs;i inalzerà da que&longs;to &longs;ito con la leua &longs;empre haurà <lb/>bi&longs;ogno di po&longs;sanza minore per e&longs;sere &longs;o&longs;tenuto: ma &longs;e &longs;arà <lb/>abba&longs;sato di maggiore. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB egualmente di&longs;tante dall'orizonte, il cui &longs;o&longs;tegno &longs;ia C, & il pe&longs;o <lb/>BD il centro della grauezza delquale &longs;ia doue è H &longs;opra la leua; & &longs;ia la po&longs;&longs;an<emph.end type="italics"/><lb/> <arrow.to.target n="fig62"></arrow.to.target><lb/><emph type="italics"/>za &longs;o&longs;tenente in A. </s> <s>Moua&longs;i dapoi la leua AB in EF, & &longs;ia il pe&longs;o mo&longs;&longs;o <lb/>in FG. </s> <s>Dico primieramente che minore po&longs;&longs;anza po&longs;ta in E &longs;o&longs;tenir à il pe&longs;o <lb/>FG con la leua EF, che la po&longs;&longs;anza in A il pe&longs;o BD con la leua AB. </s> <s>&longs;ia <lb/>il K il centro della grauezza del pe&longs;o FG. </s> <s>Dapoi &longs;iano tirate sì da H, come<emph.end type="italics"/> <pb pagenum="45"/><emph type="italics"/>da K le linee HL KM à piombo de'loro orizonti, lequali &longs;i andaranno à tro­<lb/>uare nel tentro del mondo, & &longs;ia HL à piom<gap/>o anche di e&longs;&longs;a AB. </s> <s>Dapoi &longs;ia <lb/>tirata la linea KN à piom<gap/>, di EF, laquale &longs;arà eguile ad HL, & la CN <lb/>eguale ad e&longs;&longs;a CL. </s> <s>Hor percioche HL è à piombo dell'orizonte, la po&longs;&longs;anza <lb/>in A &longs;o&longs;tenente il pe&longs;o BD haurà quella proportione ad e&longs;&longs;o pe&longs;o, che CL à<emph.end type="italics"/> <arrow.to.target n="note151"></arrow.to.target><lb/><emph type="italics"/>CA. </s> <s>Di nuouo, percioche KM è à piombo dell'orizonte, la po&longs;&longs;anza in E &longs;o­<lb/>&longs;tenente il pe&longs;o FG co&longs;i &longs;arà al pe&longs;o come CM à CE. & per e&longs;&longs;ere CN NK <lb/>eguali ad e&longs;&longs;e CL LH, & contenere angoliretti, &longs;arà CM minore di e&longs;&longs;a CL;<emph.end type="italics"/> <arrow.to.target n="note152"></arrow.to.target><lb/><emph type="italics"/>Dunque CM à CA haurà proportione minore, che CL à CA; & CA <lb/>è eguale à CE, dunque haurà CM proportione minore à CE, che CL à<emph.end type="italics"/> <arrow.to.target n="note153"></arrow.to.target><lb/><emph type="italics"/>CA: & per e&longs;&longs;erei pe&longs;i BD FG eguali, però che è il pe&longs;o mede&longs;imo. </s> <s>Dun­<lb/>que &longs;arà minore proportione della po&longs;&longs;anza in E &longs;o&longs;tenente il pe&longs;o FG ad e&longs;&longs;o <lb/>pe&longs;o, che della po&longs;&longs;anza in A &longs;o&longs;tenente il pe&longs;o BD ad e&longs;&longs;o pe&longs;o. </s> <s>Per laqual <lb/>co&longs;a minore po&longs;&longs;anza po&longs;ta in E &longs;o&longs;tenterà il pe&longs;o FG, che la po&longs;&longs;anza in A<emph.end type="italics"/> <arrow.to.target n="note154"></arrow.to.target><lb/><emph type="italics"/>il pe&longs;o BD. & quanto più &longs;arà inalzato il pe&longs;o, &longs;empre &longs;i mo&longs;trerà po&longs;&longs;anza<emph.end type="italics"/> <arrow.to.target n="note155"></arrow.to.target><lb/><emph type="italics"/>anche minore douer &longs;o&longs;tenere il pe&longs;o, per e&longs;&longs;ere la linea PC minore della CM. </s> <s><lb/>Sia dapoi la leua in QR, & il pe&longs;o in QS, il cui centro della grauezza &longs;ia O. </s> <s><lb/>Dico che po&longs;&longs;anza maggiore &longs;i richiede in R per &longs;o&longs;tenere il pe&longs;o QS, che in <lb/>A per &longs;ostentare il pe&longs;o BD. </s> <s>Tiri&longs;i dal centro O della grauezza la linea OT <lb/>a piombo dell'orizonte. </s> <s>& percioche le linee HL OT &longs;e &longs;aranno allungate dal­<lb/>la parte di L, & di T &longs;i andranno à ritrouare nel centro del mondo, &longs;arà la CT mag <lb/>giore della CL: & è la CA eguale ad e&longs;&longs;a CR, dunque la TC haurà pro-<emph.end type="italics"/> <arrow.to.target n="note156"></arrow.to.target><lb/><emph type="italics"/>portione maggiore à CR, che LC à CA. </s> <s>Maggiore dunque &longs;arà la po&longs;&longs;an-<emph.end type="italics"/> <arrow.to.target n="note157"></arrow.to.target><lb/><emph type="italics"/>za in R &longs;o&longs;tenente il pe&longs;o QS, che in A &longs;o&longs;tenente il BD. </s> <s>Similmente mo­<lb/>&longs;trera&longs;&longs;i, che quanto la leua RQ abba&longs;&longs;ando&longs;i, &longs;arà più di&longs;tante dalla leua AB,<emph.end type="italics"/> <arrow.to.target n="note158"></arrow.to.target><lb/><emph type="italics"/>&longs;empre più &longs;i ricercherà po&longs;&longs;anza mag giore à &longs;o&longs;tenere il pe&longs;o: peroche la di&longs;tanza<emph.end type="italics"/> <arrow.to.target n="note159"></arrow.to.target><lb/><emph type="italics"/>CV è più lunga di CT. </s> <s>Quanto dunque il pe&longs;o &longs;i alzerà più dal &longs;ito egualmente <lb/>di&longs;tante dall'orizonte, &longs;arà &longs;empre &longs;o&longs;tenuto da po&longs;&longs;anza minore; & quanto più &longs;i <lb/>ab<gap/>a&longs;&longs;erà, di po&longs;&longs;anza maggiore haurà me&longs;tieri per e&longs;&longs;er &longs;o&longs;tentato. </s> <s>che bi&longs;ogna­<lb/>ua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note151"></margin.target><emph type="italics"/>Per la quin ta di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note152"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note153"></margin.target><emph type="italics"/>Per la otta ua del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note154"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note155"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note156"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note157"></margin.target><emph type="italics"/>Per la otta ua del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note158"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note159"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig62"></figure> <p type="main"> <s>Quinci facilmente &longs;i caua, che la pos&longs;anza in A alla po&longs;sanza <lb/>in E co&longs;i è, come CL à CM. </s> </p> <p type="main"> <s><emph type="italics"/>Imperoche co&longs;iè LC à CA, come la po&longs;&longs;anza in A al pe&longs;o; & come CA, <lb/>cioè CE à CM, co&longs;i è il pe&longs;o alla po&longs;&longs;anza in E; Per laqual co&longs;a per la pro-<emph.end type="italics"/> <arrow.to.target n="note160"></arrow.to.target><lb/><emph type="italics"/>portion eguale, la po&longs;&longs;anza in A alla po&longs;&longs;anza in E &longs;arà come CL à CM.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note160"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 22. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Con &longs;imile ragione mo&longs;treraßi non &longs;olamente che la po&longs;&longs;anza in A co&longs;i è alla po&longs;­<lb/>&longs;anza in R, come CL à CT, ma che la po&longs;&longs;anza in E ancora alla po&longs;&longs;anza <lb/>in R è co&longs;i, come CM à CT, & co&longs;i nel re&longs;to.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia poila leua AB egualmente di&longs;tante dall'orizonte, il cui &longs;o&longs;tegno &longs;ia B, & il <lb/>centro H della grauezza del pe&longs;o CD &longs;ia &longs;opra la leua; & moua&longs;i la leua in <lb/>BE, & il pe&longs;o in FG. </s> <s>Dico che minore po&longs;&longs;anza po&longs;ta in E &longs;o&longs;tiene il pe&longs;o FG <lb/>con la leua EB, che la po&longs;&longs;anza in A il pe&longs;o CD con la leua AB. </s> <s>Sia K <lb/>il centro della grauezza del pe&longs;o FG, & da i centri delle grauezze HK &longs;iano<emph.end type="italics"/><lb/> <arrow.to.target n="fig63"></arrow.to.target><lb/> <arrow.to.target n="note161"></arrow.to.target> <emph type="italics"/>tirate le linee HL<emph.end type="italics"/> K<emph type="italics"/>M à piombo de'loro orizonti. </s> <s>Hor percioche dalle co&longs;e <lb/>di &longs;opra mo&longs;trate BM è minore di BL, & BE è eguale à BA, haurà pro-<emph.end type="italics"/><lb/> <arrow.to.target n="note162"></arrow.to.target> <emph type="italics"/>portione minore BM à BE, che BL à BA: ma come BM à BE, co&longs;i<emph.end type="italics"/><lb/> <arrow.to.target n="note163"></arrow.to.target> <emph type="italics"/>è la po&longs;&longs;anza in E &longs;o&longs;tenente il pe&longs;o FG ad e&longs;&longs;o pe&longs;o, & come BL a BA, <lb/>co&longs;i la po&longs;&longs;anza in A al pe&longs;o CD; la po&longs;&longs;anza in E al pe&longs;o FG haur à propor-<emph.end type="italics"/><lb/> <arrow.to.target n="note164"></arrow.to.target> <emph type="italics"/>tione minore, che la po&longs;&longs;anza in A al pe&longs;o CD. </s> <s>Dunque la po&longs;&longs;anza in E &longs;a­<lb/>rà minore della po&longs;&longs;anza in A. </s> <s>Similmente mo&longs;trera&longs;&longs;i quanto più il pe&longs;o &longs;i alze­<lb/>rà, &longs;empre minore po&longs;&longs;anza &longs;o&longs;tenere il pe&longs;o, ma &longs;ia la leua in BO, & il pe&longs;o in <lb/>BQ, il cui centro della grauezza &longs;ia R. </s> <s>Dico, che maggior po&longs;&longs;anza &longs;i ricerca <lb/>in O per &longs;o&longs;tenere il pe&longs;o PQ con la leua BO, che per &longs;o&longs;tenere il pe&longs;o CD con <lb/>la leua BA. </s> <s>Sia tirata dal punto R la linea RS à piombo dell'orizonte. </s> <s>&<emph.end type="italics"/><lb/> <arrow.to.target n="note165"></arrow.to.target> <emph type="italics"/>percioche BS è maggiore di BL, haurà BS proportione maggiore à BO, che <lb/>BL à BA; Per laqual co&longs;a la po&longs;&longs;anza in O &longs;o&longs;tenente il pe&longs;o PQ &longs;arà maggio <lb/>re della po&longs;&longs;anza in A &longs;o&longs;tenente il pe&longs;o CD. & à que&longs;to modo &longs;i mo&longs;trerà an­<lb/>cora che quanto la leua BO abba&longs;&longs;an lo&longs;i, &longs;arà più di&longs;tante dalla leua AB &longs;em­<lb/>pre vi vorrà po&longs;&longs;anza maggiore à &longs;o&longs;iener il pe&longs;o.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note161"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note162"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note163"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note164"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note165"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig63"></figure> <p type="main"> <s><emph type="italics"/>Di qui parimente, come di &longs;opra è mani&longs;e&longs;to, che la po&longs;&longs;anza in A è alla po&longs;&longs;anza in<emph.end type="italics"/> <pb pagenum="46"/><emph type="italics"/>B, come BL à BM: & la po&longs;&longs;anza in A alla po&longs;&longs;anza in O, come BL à BS. & <lb/>la po&longs;&longs;anza in E alla po&longs;&longs;anza in O, come BM à BS.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Oltre à ciò &longs;e &longs;i intenderà vn'altra po&longs;&longs;anza in B, per modo che due &longs;iano le po&longs;&longs;an­<lb/>ze, che &longs;o&longs;tentino il pe&longs;o, minore &longs;arà la po&longs;&longs;anza in B, che &longs;o&longs;tiene il pe&longs;o PQ <lb/>con la leua BO, che il pe&longs;o CD con la leua BA. ma per lo contrario &longs;i ri­<lb/>cerca po&longs;&longs;anza maggiore in B per &longs;o&longs;tenere il pe&longs;o FG con la leua BE, che <lb/>il pe&longs;o CD con la leua AB: percioche tirata la linea KN à piombo di EB, <lb/>&longs;arà EN eguale ad AL: Per laqual co&longs;a EM &longs;arà maggiore di LA. </s> <s>Dun<emph.end type="italics"/> <arrow.to.target n="note166"></arrow.to.target><lb/><emph type="italics"/>que EM haurà proportione maggiore ad EB, che LA ad AB, & LA <lb/>maggiore ad AB, che SO ad OB, lequali &longs;ono proportioni della po&longs;&longs;anza<emph.end type="italics"/> <arrow.to.target n="note167"></arrow.to.target><lb/><emph type="italics"/>al pë&longs;o.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note166"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note167"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Similmente prouera&longs;&longs;i, che la po&longs;&longs;anza in B &longs;o&longs;tenente il pe&longs;o con la leua AB è al­<lb/>la po&longs;&longs;anza &longs;o&longs;tenente po&longs;ta nell'i&longs;te&longs;&longs;o punto B con la leua EB, come LA <lb/>ad EM; & co&longs;i e&longs;&longs;ere anche alla po&longs;&longs;anza di B &longs;o&longs;tenente il pe&longs;o con la leua OB, <lb/>come AL ad OS. </s> <s>Ma quelle po&longs;&longs;anze che &longs;o&longs;tengono con le leue EB OB <lb/>&longs;ono co&longs;itraloro come EM ad OS.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Dapoi mo&longs;treremo come nelle co&longs;e che di &longs;opra &longs;ono &longs;tate dette, che la po&longs;&longs;anza in B <lb/>ha quella proportione alla po&longs;&longs;anza in E, che EM ad MB; & la po&longs;&longs;anza<emph.end type="italics"/> <arrow.to.target n="note168"></arrow.to.target><lb/><emph type="italics"/>in B co&longs;i e&longs;&longs;ere alla po&longs;&longs;anza in A, come AL ad LB, & la po&longs;&longs;anza in B <lb/>alla po&longs;&longs;anza in O, come OS ad SB.<emph.end type="italics"/> <arrow.to.target n="note169"></arrow.to.target></s> </p> <p type="margin"> <s><margin.target id="note168"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 3. <emph type="italics"/>ce roll ario.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note169"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma &longs;ia la leua AB e­<lb/>gualmente di&longs;tante <lb/>dall'orizonte, il cui <lb/>&longs;o&longs;tegno &longs;ia B, & <lb/>il centro H della <lb/>grauezza del pe&longs;o <lb/>AC &longs;ia &longs;opra la <lb/>leua: & moua&longs;i la <lb/>leua in BE, & il <lb/>pe&longs;o in EF, & la <lb/>po&longs;&longs;anza in G. di <lb/>mo&longs;trera&longs;&longs;i parimen <lb/>te, come di &longs;opra, che <lb/>la po&longs;&longs;anza in G &longs;o <lb/>&longs;tenente il pe&longs;o EF <lb/>è minore della po&longs;­<lb/>&longs;anza in D &longs;o&longs;te-<emph.end type="italics"/><lb/> <arrow.to.target n="fig64"></arrow.to.target><lb/><emph type="italics"/>nente il pe&longs;o AC. percioche e&longs;&longs;endo minore BM di BL haurà minore pro­<lb/>portione MB à BG, che LB à BD. & à que&longs;to modo proueraßi, che quan <lb/>to il pe&longs;o più &longs;i alzerà con la leua, &longs;empre minore po&longs;&longs;anza &longs;i ricerca à &longs;o&longs;tenere<emph.end type="italics"/> <pb/><emph type="italics"/>il detto pe&longs;o. </s> <s>&longs;imilmente &longs;e la leua &longs;i moue in BO, & la po&longs;&longs;anza &longs;o&longs;tenente &longs;ia <lb/>in N, &longs;i mo&longs;trerà <lb/>la po&longs;&longs;anza in N e&longs;­<lb/>&longs;ere maggiore della <lb/>po&longs;&longs;anza in D. pe­<lb/>roche SB ha pro­<lb/>portione maggiore <lb/>à BN che LB <lb/>à BD. </s> <s>Mo&longs;tre­<lb/>raßi ancora, che <lb/>quanto il pe&longs;o più <lb/>s'abb a&longs;&longs;erà, &longs;empre <lb/>ricercar&longs;i po&longs;&longs;anza <lb/>maggiore à &longs;o&longs;tene­<lb/>re il pe&longs;o. </s> <s>che bi&longs;o­<lb/>gnaua mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig64"></figure> <figure></figure> <p type="main"> <s><emph type="italics"/>Di quì <expan abbr="parimēte">parimente</expan> è chia <lb/>ro, che le po&longs;&longs;anze <lb/>in GDN co&longs;itraloro &longs;ono, come BM à BL, & come BL à BS, & vlti­<lb/>mamente come BM à BS.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO</s> </p> <p type="main"> <s>Da que&longs;te co&longs;e è manife&longs;to, che &longs;e la po&longs;sanza con la leua moue <lb/>rà in sù il pe&longs;o, il cui centro della grauezza &longs;ia &longs;opra la leua, <lb/>quanto più &longs;arà alzato il pe&longs;o, &longs;empre vi vorrà po&longs;sanza mi­<lb/>nore per mouere il pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche doue la po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o è &longs;empre minore, &longs;arà parimente la po&longs;­<lb/>&longs;anza, che lo moue &longs;empre minore.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Da que&longs;te co&longs;e dimo&longs;treraßi etiandio, &longs;ia pur il centro della grauezza del pe&longs;o mede&longs;i­<lb/>mo ò più da pre&longs;&longs;o, ò più da lunge della leua AB egualmente di&longs;tante dall' ori­<lb/>zonte, che la po&longs;&longs;anza mede&longs;ima in A &longs;o&longs;terrà nondimeno il pe&longs;o: come &longs;e il cen <lb/>tro H della grauezza del pe&longs;o BD &longs;ia più da lunge dalla leua BA, che il cen­<lb/>tro N della grauezza del pe&longs;o PV, pur che la linea HL tirata dal punto H <lb/>à piombo dell'orizonte, & della leua AB paßi per N, & &longs;ia il pe&longs;o PV <lb/>eguale al pe&longs;o BD; &longs;arà sì il pe&longs;o BD, & sì il pe&longs;o PV come &longs;e ambidue &longs;o&longs;­<lb/>&longs;ero appiccati ad L; & &longs;ono eguali per e&longs;&longs;ere pre&longs;i in luogo di vn pe&longs;o &longs;olo, dun­<lb/>que la i&longs;te&longs;&longs;a po&longs;&longs;anza in A &longs;o&longs;ienente il pe&longs;o BD &longs;o&longs;terrà anche il pe&longs;o PV.<emph.end type="italics"/> <pb pagenum="47"/><emph type="italics"/>Ma nella leua EF quanto il centro della grauezza &longs;arà più da lunge dalla leua. </s> <s><lb/>tanto più egualmente la po&longs;&longs;anza &longs;o&longs;tenter à il pe&longs;o mede&longs;imo, come &longs;e il centro K <lb/>della grauezza del pe&longs;o FG &longs;o&longs;&longs;e più da lunge dalla leua EF, che il centro X <lb/>dalla grauezza del pe&longs;o <foreign lang="greek">*u</foreign>Z; in modo però, che la lineatirata dal punto<emph.end type="italics"/> K <emph type="italics"/>à <lb/>piombo della leua FE paßi per X; & &longs;ia il pe&longs;o FG eguale al pe&longs;o <foreign lang="greek">*u</foreign>Z; <lb/>& da punti KX &longs;iano tirate le linee KM X<36> à piombo de loro orizonti; &longs;a-<emph.end type="italics"/><lb/> <arrow.to.target n="fig65"></arrow.to.target><lb/><emph type="italics"/>rà la C<36> maggiore di CM; & perciò il pe&longs;o FG &longs;arà nella leua co&longs;i come <lb/>&longs;e fo&longs;&longs;e appiccato in M, & il pe&longs;o <foreign lang="greek">*u</foreign>Z come fo&longs;&longs;e appiccato in <36>. </s> <s>Hor per-<emph.end type="italics"/> <arrow.to.target n="note170"></arrow.to.target><lb/><emph type="italics"/>cioche C<36> ha proportione maggiore à CE, che CM à CE, maggiore <lb/>&longs;arà la po&longs;&longs;anza po&longs;ta in E, che &longs;o&longs;terrà il pe&longs;o <foreign lang="greek">*u</foreign>Z, che FG. </s> <s>Manella leua <lb/>QR per lo contrario &longs;i dimo&longs;trerà, cioè che quanto il centro della grauezza del pe <lb/>&longs;o mede&longs;imo è più da lunge dalla leua, tanto più anche maggiore è la po&longs;&longs;anza che <lb/>&longs;o&longs;tiene il pe&longs;o. </s> <s>peroche maggiore è CT di CI, & perciò CT hauerà proportio­<lb/>ne maggiore à CR, che CI à CR. </s> <s>&longs;imilmente dimo&longs;treraßi, &longs;e il pe&longs;o &longs;arà col <lb/>locato fra la po&longs;&longs;anza, & il &longs;o&longs;tegno, ouero la po&longs;&longs;anza po&longs;ta fra il &longs;o&longs;tegno, & il <lb/>pe&longs;o, il che mede&longs;imamente auuenirà alla po&longs;&longs;anzà che moue peroche doue po&longs;&longs;anza <lb/>minore &longs;o&longs;tiene il pe&longs;o, iui po&longs;&longs;anza minore lo mouerà: & doue &longs;iricerca po&longs;&longs;anza <lb/>maggiore in &longs;o&longs;tenere, iui anche maggiore vi vuole in mouere.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note170"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure id="fig65"></figure> <p type="head"> <s>PROPOSITIONE IX.</s> </p> <p type="main"> <s>La po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o, che habbia il centro della &longs;ua gra <lb/>uezza &longs;otto la leua egualmente di&longs;tante dall'orizonte, quanto <lb/>più il pe&longs;o &longs;arà alzato da que&longs;to &longs;ito con la leua, haurà egli &longs;em <lb/>preanco me&longs;tieri di po&longs;&longs;anza maggiore ad e&longs;&longs;ere &longs;o&longs;tenuto; <lb/>Ma &longs;e abba&longs;&longs;ato, di minore. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB egualmente <expan abbr="di&longs;tāte">di&longs;tante</expan> dall'orizonte, il cui &longs;o&longs;tegno &longs;ia C, & &longs;ia il pe&longs;o AD, <lb/>il cui centro L della grauezza &longs;ia &longs;otto la leua, & &longs;ia in B la po&longs;&longs;anza &longs;o&longs;tenen­<lb/>te il pe&longs;o AD: moua&longs;i dopo la leua in FG, & il pe&longs;o in FH. </s> <s>Dico prima, <lb/>che po&longs;&longs;anza maggiore &longs;i ricerca in G per &longs;o&longs;tenere il pe&longs;o FH con la leua FG, <lb/>di quel che &longs;iala po&longs;&longs;anza in B e&longs;&longs;endo il pe&longs;o AD, ma con la leua AB. </s> <s>&longs;ia<emph.end type="italics"/><lb/> <arrow.to.target n="fig66"></arrow.to.target><lb/><emph type="italics"/>M il centro della grauezza del pe&longs;o FH, & da punti LM &longs;iano tirate le linee<emph.end type="italics"/><lb/> <arrow.to.target n="note171"></arrow.to.target> <emph type="italics"/>LK MN à piombo de'loro orizonti; & &longs;iatirata la linea MS à piombo di FG, <lb/>che &longs;arà eguale ad LK, & CK &longs;arà etiandio eguale ad e&longs;&longs;a CS. </s> <s>Percioche dun<emph.end type="italics"/><lb/> <arrow.to.target n="note172"></arrow.to.target> <emph type="italics"/>que CN è maggiore di CK haurà NC proportione maggiore à CG, che CK <lb/>à CB; & la po&longs;&longs;anza in B al pe&longs;o AD ha la mede&longs;ma proportione, che KC<emph.end type="italics"/><lb/> <arrow.to.target n="note173"></arrow.to.target> <emph type="italics"/>à CB: & come la po&longs;&longs;anza in G al pe&longs;o FH, co&longs;i è NC à CG; dunque la<emph.end type="italics"/><lb/> <arrow.to.target n="note174"></arrow.to.target> <emph type="italics"/>po&longs;&longs;anza in G hauer à maggiore proportione al pe&longs;o FH, che la po&longs;&longs;anza in B <lb/>al pe&longs;o AD. </s> <s>Maggiore dunque è la po&longs;&longs;anza in G della po&longs;&longs;anza in B. che&longs;e<emph.end type="italics"/> <pb pagenum="48"/><emph type="italics"/>la leua &longs;arà in OP, & il pe&longs;o in <expan abbr="Oq;">Oque</expan> &longs;arà la po&longs;&longs;anza po&longs;ta in B maggiore, <lb/>che in P: percioche &longs;i dimo&longs;trerà nell'i&longs;te&longs;&longs;o modo CR e&longs;&longs;ere minore di CK, & <lb/>CR hauere proportione minore a CP, che CK a CB; & perciò la po&longs;&longs;anza <lb/>po&longs;ta in B e&longs;&longs;ere maggiore della po&longs;&longs;anza po&longs;ta in P. </s> <s>& a que&longs;to modo mo&longs;tre­<lb/>ra&longs;&longs;i che quanto più il pe&longs;o &longs;i alzerà dal &longs;ito AB, &longs;empre vi vorrà po&longs;&longs;anza mag­<lb/>giore à &longs;o&longs;tenerlo. </s> <s>ma per lo contrario accaderà &longs;e egli &longs;arà abba&longs;&longs;ato. </s> <s>che bi&longs;o­<lb/>gnaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note171"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>di qu sto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note172"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note173"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note174"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure id="fig66"></figure> <p type="main"> <s><emph type="italics"/>Di quà ancora &longs;i puote ageuolmente cauare, che le po&longs;&longs;anze po&longs;te in PBG &longs;ono in <lb/>modo di&longs;po&longs;te fraloro, come CR à CK; & come CK à CN, & come CN <lb/>à CR.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Sia dopo la leua AB egualmente di&longs;tante dall'orizonte, co'l &longs;uo &longs;o&longs;tegno B; & il <lb/>pe&longs;o CD habbia il centro O della grauezza &longs;otto la leua, & &longs;ia in A la po&longs;­<lb/>&longs;anza &longs;o&longs;tenente il pe&longs;o CD. </s> <s>Moua&longs;i dapoi la leua in BE, & BF, & &longs;i tra­<lb/>&longs;porti il pe&longs;o in GH KL. </s> <s>Dico, che maggiore po&longs;&longs;anza per &longs;o&longs;tenere il pe&longs;o &longs;i<emph.end type="italics"/><lb/> <arrow.to.target n="fig67"></arrow.to.target><lb/><emph type="italics"/>ricerca in E, che in A; & maggiore in A che in F &longs;iano tirate da i centri <lb/>delle grauezze le linee NM OP QR à piombo de gli orizonti, lequali allun <lb/>gate da la parte di NOQ &longs;i andranno à trouare nel centro del mondo. </s> <s>Mo&longs;tre­<lb/>ra&longs;&longs;i parimente come di &longs;opra, che BM è maggiore di BP, & BP maggio-<emph.end type="italics"/> <arrow.to.target n="note175"></arrow.to.target><lb/><emph type="italics"/>re di BR; & che BM ha proportione maggiore à BE, che BP à BA; & <lb/>BP à BA maggiore che BR à BF: & per que&longs;to la po&longs;&longs;anza in E mag­<lb/>giore è della po&longs;&longs;anza in A; & la po&longs;&longs;anza in A maggiore della po&longs;&longs;anza in<emph.end type="italics"/><lb/>F. <emph type="italics"/>& quanto la leua &longs;i alzerà più dal &longs;ito AB, mo&longs;trera&longs;&longs;i &longs;empre, che mag- <pb/>giore po&longs;&longs;anza vi vuole à &longs;o&longs;tenere il pe&longs;o: ma &longs;e abba&longs;&longs;eraßi, minore.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note175"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig67"></figure> <p type="main"> <s><emph type="italics"/>Di quì è chiaro etiandio che le po&longs;&longs;anze po&longs;te in EAF co&longs;itraloro &longs;ono, come BM <lb/>à BP, & come BP à BR, & come BM à BR.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Di più &longs;e in B &longs;arà vn'altra po&longs;&longs;anza, per modo, che due po&longs;&longs;anze &longs;iano quelle che <lb/>&longs;o&longs;tengano il pe&longs;o. </s> <s>Di maggiore po&longs;&longs;anza è bi&longs;ogno in B per &longs;o&longs;tenere il pe&longs;o KL <lb/>con la leua BF, che per &longs;o&longs;tenere il pe&longs;o CD con la leua AB. & dauan-<emph.end type="italics"/><lb/> <arrow.to.target n="fig68"></arrow.to.target><lb/><emph type="italics"/>taggio anco maggiore con la leua AB, che con la leua BE: peroche RF ha <lb/>proportione maggiore ad FB, che PA ad AB; & PA ad AB mag­<lb/>giore, che EM ad EB.<emph.end type="italics"/></s> </p> <figure id="fig68"></figure> <p type="main"> <s><emph type="italics"/>Similmente mo&longs;treraßi, che le po&longs;&longs;anze in B &longs;o&longs;tenenti il pe&longs;o con le leue traloro co&longs;i <lb/>e&longs;&longs;ere, come EM ad AP, & come AP ad FR, & come EM ad FR.<emph.end type="italics"/></s> </p> <p type="main"> <s> <arrow.to.target n="note176"></arrow.to.target> <emph type="italics"/>Oltre à ciò la po&longs;&longs;anza in B co&longs;i &longs;arà alla po&longs;&longs;anza in F, come RF ad RB; & <lb/>la po&longs;&longs;anza in B alla po&longs;&longs;anza in A come PA à PB, & la po&longs;&longs;anza in B al-<emph.end type="italics"/><lb/> <arrow.to.target n="note177"></arrow.to.target> <emph type="italics"/>la po&longs;&longs;anza in E come EM ad MB.<emph.end type="italics"/></s> </p> <pb pagenum="49"/> <p type="margin"> <s><margin.target id="note176"></margin.target><emph type="italics"/>Per lo<emph.end type="italics"/> 3. <emph type="italics"/>co<gap/>llario.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note177"></margin.target><emph type="italics"/><gap/>er la<emph.end type="italics"/> 2. <emph type="italics"/>di <gap/>esto.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma&longs;ia la leua AB egualmente di&longs;tante <expan abbr="dall'orizõte">dall'orizonte</expan>, col &longs;uo &longs;o&longs;tegno B, & il pe&longs;o AC, <lb/>il cui centro della <lb/>grauezza &longs;ia &longs;ot­<lb/>to la leua, & &longs;ia <lb/>la po&longs;&longs;anza <expan abbr="&longs;o&longs;te-nēce">&longs;o&longs;te­<lb/>nence</expan> il pe&longs;o in D, <lb/>& moua&longs;i la le­<lb/>ua in BE BF, <lb/>& la po&longs;&longs;anza <lb/>in GH; &longs;imil­<lb/>mente mo&longs;trera&longs; <lb/>&longs;i, che la po&longs;&longs;an­<lb/>za in G è mag <lb/>giore della po&longs;&longs;an <lb/>za in D, & la <lb/>po&longs;&longs;anza in D <lb/>maggiore della<emph.end type="italics"/><lb/> <arrow.to.target n="fig69"></arrow.to.target><lb/><emph type="italics"/>po&longs;&longs;anza in H. percioche KB ha proportione maggiore à BG, che BL à BD, <lb/>& BL à BD maggiore che MB à BH. </s> <s>& à questa maniera mo&longs;trera&longs;&longs;i che <lb/>quanto la leua più &longs;i alzerà dal &longs;ito AB, dauantaggio douere &longs;empre e&longs;&longs;ere mag <lb/>gior la po&longs;&longs;anza per &longs;o&longs;tenere il pe&longs;o: & quanto più s'abba&longs;&longs;a, minore. </s> <s>che dimo <lb/>&longs;trare era me&longs;tieri.<emph.end type="italics"/></s> </p> <figure id="fig69"></figure> <p type="main"> <s><emph type="italics"/>Similmente in que&longs;te, le po&longs;&longs;anze poste in GDH co&longs;itraloro &longs;aranno, come BK à <lb/>BL, & come BL à BM, & alla &longs;ine come BK à BM.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Da que&longs;te co&longs;e etiandio è pale&longs;e, che &longs;e la po&longs;&longs;anza mouerà con <lb/>la leua in sù vn pe&longs;o, che habbia il centro della grauezza &longs;otto <lb/>la leua; Quanto più il pe&longs;o &longs;arà alzato, &longs;empre vi vorrà po&longs;­<lb/>&longs;anza maggiore per mouere il pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Imperoche &longs;e la po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o è &longs;empre maggiore, &longs;arà parimente la <lb/>po&longs;&longs;anza che moue il pe&longs;o &longs;empre maggiore.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Da que&longs;te co&longs;e anco &longs;i cauerà facilmente &longs;e &longs;arà il centro della grauezza dell'i&longs;te&longs;&longs;o pe <lb/>&longs;o ò più da pre&longs;&longs;o, ò più da lunge dalla leua AB egualmente di&longs;tante dall'orizon <lb/>te, che la po&longs;&longs;anza mede&longs;ima po&longs;ta in B &longs;o&longs;terrà il pe&longs;o. </s> <s>come &longs;e il centro L della <lb/>grauezza del pe&longs;o AD fo&longs;&longs;e più da lunge dalla leua BA, che il centro N <lb/>della grauezza del pe&longs;o PV, pur che la linea LK tirata dal punto L à piom <lb/>bo dell orizonte, & della leua AB pa&longs;&longs;i per N: &longs;imilmente come nella prece-<emph.end type="italics"/><lb/> <arrow.to.target n="fig70"></arrow.to.target><lb/><emph type="italics"/>dente &longs;i mo&longs;trerà, che la po&longs;&longs;anza mede&longs;ima in B &longs;ostiene & il pe&longs;o AD, & <lb/>il pe&longs;o PV. </s> <s>Ma nella leua EF quanto il centro della grauezza &longs;arà più da lun <lb/>ge dalla leua, tanto haur à me&longs;tieri di po&longs;&longs;anza maggiore per &longs;ostenere il pe&longs;o. </s> <s>co­<lb/>me il centro M della grauezza del pe&longs;o FH &longs;ia più da lunge dalla leua EF, che <lb/>il centro S della grauezza del pe&longs;o XZ. </s> <s>&longs;iano tirate da i punti MS le linee <lb/>MI SG à piombo de gli orizonti; &longs;arà CI maggiore di CG: & perciò la po&longs;&longs;an <lb/>za di E deue e&longs;&longs;ere maggiore &longs;o&longs;tenendo il pe&longs;o FH, che il pe&longs;o XZ. </s> <s>Maper <lb/>lo contrario &longs;i mo&longs;trerà nella leua OR, cioè che quanto il centro della grauezza <lb/>dell'i&longs;te&longs;&longs;o pe&longs;o è più dalunge dalla leua, il pe&longs;o viene &longs;o&longs;tentato da po&longs;&longs;anza mino <lb/>re. </s> <s>peroche minore è C<foreign lang="greek">*u</foreign> de CT. </s> <s>& in modo &longs;imile demo&longs;traraßi ancora &longs;tan <lb/>do il pe&longs;o fra la po&longs;&longs;anza, & il &longs;o&longs;tegno, ouero la po&longs;&longs;anzatra il &longs;ostegno, & il<emph.end type="italics"/> <pb pagenum="50"/><emph type="italics"/>pe&longs;o, ilche parimente auerrà alla po&longs;&longs;anza che moue; peroche doue po&longs;&longs;anza mino­<lb/>re &longs;o&longs;tien il pe&longs;o, iui minore po&longs;&longs;anza lo mouerà. </s> <s>& doue vuole po&longs;&longs;anza maggio­<lb/>re in &longs;o&longs;tentare, iui anco ella &longs;arà maggiore in mouere.<emph.end type="italics"/></s> </p> <figure id="fig70"></figure> <p type="head"> <s>PROPOSITIO NE X.</s> </p> <p type="main"> <s>La po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o che habbia il centro della grauez­<lb/>za nella i&longs;te&longs;&longs;a leua, &longs;ia pure in qual &longs;i voglia modo tra&longs;porta­<lb/>to il pe&longs;o con la leua; vi &longs;arà &longs;empre me&longs;tieri della po&longs;&longs;anza <lb/>i&longs;te&longs;&longs;a, acciò &longs;ia &longs;o&longs;tenuto. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB egualmente di&longs;tante dall'orizonte, co'l luo &longs;o&longs;tegno C, & E cen­<lb/>tro della grauezza del pe&longs;o &longs;ia in e&longs;&longs;a leua. </s> <s>Moua&longs;i dapoi la leua in FG, & HK,<emph.end type="italics"/><lb/> <arrow.to.target n="fig71"></arrow.to.target><lb/><emph type="italics"/>& il centro della grauezza in LM. </s> <s>Dico che la mede&longs;ima po&longs;&longs;anza di KBG &longs;em-<emph.end type="italics"/> <arrow.to.target n="note178"></arrow.to.target><lb/><emph type="italics"/>pre &longs;o&longs;terrà l'iste&longs;&longs;o pe&longs;o. </s> <s>Hor percioche il pe&longs;o nella leua AB è &longs;i fattamen­<lb/>te di&longs;po&longs;to, come &longs;e egli fo&longs;&longs;e appiccato in E; & nella leua GF come &longs;e eglifo&longs; <lb/>&longs;e appiccato in L; & nella leua HK, come &longs;e egli fo&longs;&longs;e appiccato in M; & le<emph.end type="italics"/> <pb/><emph type="italics"/>distanze CL CE CM &longs;ono traloro eguali; & parimente CK CB CG pur <lb/>tra loro eguali; &longs;arà la po&longs;&longs;anza in B al pe&longs;o, come CE à CB; & la po&longs;&longs;an­<lb/>za in K al pe&longs;o, come CM à CK, & la po&longs;&longs;anza in G al pe&longs;o, come CL<emph.end type="italics"/><lb/> <arrow.to.target n="fig72"></arrow.to.target><lb/><emph type="italics"/>à CG. </s> <s>La po&longs;&longs;anza mede&longs;ma dunque in KBG &longs;osterrà il pe&longs;o mede&longs;mo tra&longs;por <lb/>tato in vari &longs;iti. </s> <s>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note178"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig71"></figure> <figure id="fig72"></figure> <p type="main"> <s><emph type="italics"/>Similmente prouera&longs;&longs;i, &longs;e il pe&longs;o fo&longs;&longs;e tra la po&longs;&longs;anza, & il &longs;o&longs;tegno; ouero la po&longs;­<lb/>&longs;anza tra il &longs;o&longs;tegno, & il pe&longs;o, che il mede&longs;imo auerrà alla po&longs;&longs;anza, che moue.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE XI.</s> </p> <p type="main"> <s>Se la di&longs;tanza della leua tra il &longs;o&longs;tegno, & la po&longs;&longs;anza haurà pro­<lb/>portione maggiore alla di&longs;tanza trapo&longs;ta dal &longs;o&longs;tegno al pun­<lb/>to, doue dal centro della grauezza del pe&longs;o tirata vna linea à <lb/>piombo dell'orizonte taglia la leua che non ha il pe&longs;o alla po&longs; <lb/>&longs;anza; il pe&longs;o veramente &longs;arà mo&longs;&longs;o dalla po&longs;&longs;anza. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB, & dal punto A appicchi&longs;i il pe&longs;o C; cioè il punto A &longs;empre <lb/>&longs;ia quel punto, doue la linea tirata à piombo dal centro della grauezza del pe&longs;o ta­<lb/>gli la leua; & &longs;ia la po&longs;&longs;anza in B, & il &longs;ostegno D; & DB habbia à DA<emph.end type="italics"/> <pb pagenum="51"/><emph type="italics"/>proportione maggiore, che il pe&longs;o C alla po&longs;&longs;anza in B. </s> <s>Dico che il pe&longs;o C &longs;a­<lb/>rà mo&longs;&longs;o dalla po&longs;&longs;anza in B. </s> <s>Faccia&longs;i come BD à DA, co&longs;i il pe&longs;o E alla<emph.end type="italics"/> <arrow.to.target n="note179"></arrow.to.target><lb/><emph type="italics"/>po&longs;&longs;anza in B; & appicchi&longs;i parimente il pe&longs;o E in A: egliè chiaro che la po&longs;­<lb/>&longs;anza in B pe­<lb/>&longs;a <expan abbr="egualmēte">egualmente</expan> <expan abbr="cõ">com</expan> <lb/>e&longs;&longs;o E; cioè che <lb/>&longs;o&longs;tiene il detto <lb/>pe&longs;o E. </s> <s>& per­<lb/>cioche BD ha <lb/>proportion mag <lb/>giore à DA che <lb/>C alla po&longs;&longs;anza <lb/>in B. & come <lb/>BD à DA, co&longs;i<emph.end type="italics"/><lb/> <arrow.to.target n="fig73"></arrow.to.target><lb/><emph type="italics"/>è il pe&longs;o F. alla po&longs;&longs;anza: adunque E haurà proportione maggiore alla po&longs;&longs;an-<emph.end type="italics"/> <arrow.to.target n="note180"></arrow.to.target><lb/><emph type="italics"/>za, che il pe&longs;o C alla po&longs;&longs;anza i&longs;te&longs;&longs;a. </s> <s>Per laqual co&longs;a il pe&longs;o E &longs;arà maggiore <lb/>del pe&longs;o C. & perche la po&longs;&longs;anza pe&longs;a egualmente cone&longs;&longs;o E; dunque la po&longs;&longs;an <lb/>za non pe&longs;erà egualmente cone&longs;&longs;o C, ma per la forza &longs;ua inchinerà al ba&longs;&longs;o. </s> <s>àun <lb/>que il pe&longs;o C &longs;arà mo&longs;&longs;o dalla po&longs;&longs;anza in B con la leua AB, il cui &longs;o&longs;tegno <lb/>è in D.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note179"></margin.target><emph type="italics"/>Per la prima di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note180"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure id="fig73"></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e la leua fo&longs;&longs;e AB, & il &longs;o&longs;tegno A, & il pe&longs;o C appiccato in D, & la <lb/>po&longs;&longs;anza in B, & BA haue&longs;&longs;e proportione maggiore ad AD, che il pe&longs;o C <lb/>alla po&longs;&longs;anza in B. </s> <s>Dico che il pe&longs;o C mouera&longs;&longs;i dalla po&longs;&longs;anza in B. </s> <s>faccia&longs;i co<emph.end type="italics"/> <arrow.to.target n="note181"></arrow.to.target><lb/><emph type="italics"/>me BA ad AD, co&longs;i il pe­<lb/>&longs;o E alla po&longs;&longs;anza in B: & <lb/>&longs;e E &longs;arà appiccato in D, la <lb/>po&longs;&longs;anza in B &longs;o&longs;tenterà il pe­<lb/>&longs;o E. </s> <s>Ma per hauere BA pro­<lb/>portione maggiore ad AD, <lb/>che il pe&longs;o C alla po&longs;&longs;anza in <lb/>B; & come BA ad AD, <lb/>co&longs;i è il pe&longs;o E alla po&longs;&longs;anza in <lb/>B; dunque il pe&longs;o E haurà pro <lb/>portione maggiore alla po&longs;&longs;an<emph.end type="italics"/> <arrow.to.target n="note182"></arrow.to.target><lb/> <arrow.to.target n="fig74"></arrow.to.target><lb/><emph type="italics"/>za che è in B, che il pe&longs;o C all'i&longs;te&longs;&longs;a po&longs;&longs;anza: & perciò il pe&longs;o E &longs;arà maggio <lb/>re del pe&longs;o C; & la po&longs;&longs;anza in B &longs;o&longs;tiene il pe&longs;o E; dunque la po&longs;&longs;anza in B <lb/>con la leua AB mouerà il pe&longs;o C minore del pe&longs;o E appiccato in D, il cui &longs;o­<lb/>stegno è A.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note181"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note182"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <figure id="fig74"></figure> <p type="main"> <s><emph type="italics"/>Sia da capo la leua AB, & il &longs;uo &longs;o&longs;tegno A, & il pe&longs;o C &longs;ia appiccato in B, <lb/>& &longs;ia la po&longs;&longs;anza in D: & DA habbia proportione maggiore ad AB, che <lb/>il pe&longs;o C al­<lb/>la po&longs;&longs;anza, <lb/>che è in D. </s> <s>Di <lb/>co che il pe&longs;o C <lb/>&longs;arà mo&longs;&longs;o dal <lb/>la <expan abbr="poßāza">poßanza</expan> che <lb/>è in D. </s> <s>Fac­<lb/>cia&longs;i come D <lb/>A ad AB, <lb/>co&longs;i il pe&longs;o E <lb/>alla po&longs;&longs;anza,<emph.end type="italics"/><lb/> <arrow.to.target n="fig75"></arrow.to.target><lb/><emph type="italics"/>che è in D; & &longs;ia il pe&longs;o E pendente dal punto B: la po&longs;&longs;anza in D &longs;o&longs;ter­<lb/>rà il pe&longs;o E. </s> <s>Ma DA tiene proportione maggiore ad AB, che C alla po&longs;­<lb/>&longs;anza in D. & come DA ad AB, co&longs;i è il pe&longs;o E alla po&longs;&longs;anza in D; <lb/>dunque il pe&longs;o E haurà proportione maggiore alla po&longs;&longs;anza che è in D, che il <lb/>pe&longs;o C alla i&longs;te&longs;&longs;a po&longs;&longs;anza. </s> <s>Per laqual co&longs;a il pe&longs;o E è maggiore del pe&longs;o C. </s> <s><lb/>Et percioche la po&longs;&longs;anza in D &longs;o&longs;tiene il pe&longs;o E, dunque la detta po&longs;&longs;anza in <lb/>D mouerà il pe&longs;o C appiccato in B con la leua AB, il cui &longs;o&longs;tegno è A. che <lb/>bi&longs;ognaua prouare.<emph.end type="italics"/></s> </p> <figure id="fig75"></figure> <p type="head"> <s>Altramente.</s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB, & il pe&longs;o C appiccato in A, & la po&longs;&longs;anza in B, & &longs;ia il <lb/>&longs;o&longs;tegno D; & DB habbia proportione maggiore à DA, che il pe&longs;o C alla <lb/>po&longs;&longs;anza in B. </s> <s><lb/>Dico che il pe­<lb/>&longs;o C &longs;arà mo&longs; <lb/>&longs;o dalla po&longs;&longs;an­<lb/>za in B. </s> <s>Fac­<lb/>cia&longs;i BE ad <lb/>EA, come il<emph.end type="italics"/><lb/> <arrow.to.target n="fig76"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o C &longs;i ha inuer&longs;o la po&longs;&longs;anza, &longs;arà il punto E tra BD: percioche egli è me-<emph.end type="italics"/><lb/> <arrow.to.target n="note183"></arrow.to.target> <emph type="italics"/>&longs;tieri che BE habbia proportione minore ad EA, che DB à DA; & però <lb/>BE &longs;arà minore di BD. & percioche la po&longs;&longs;anza in B &longs;o&longs;tiene il pe&longs;o C ap­<lb/>piccato in A con la leua AB, che hà il &longs;o&longs;tegno E; dunque minore po&longs;&longs;an­<lb/>za po&longs;ta in B, che la data &longs;o&longs;terrà il pe&longs;o mede&longs;imo nel &longs;o&longs;tegno D. </s> <s>La po&longs;&longs;an­<lb/>za data dunque po&longs;ta in B mouerà il pe&longs;o C con la leua AB, che ha il &longs;o&longs;te­<lb/>gno in D.<emph.end type="italics"/></s> </p> <pb pagenum="52"/> <p type="margin"> <s><margin.target id="note183"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig76"></figure> <p type="main"> <s><emph type="italics"/>Sia dapoila leua AB, & il &longs;uo &longs;o&longs;tegno in A, & il pe&longs;o C appiccato in D, & <lb/>&longs;iala po&longs;&longs;anza in B; & AB habbia proportione maggiore ad AD, che il <lb/>pe&longs;o C alla po&longs;&longs;anza in B. </s> <s>Di <lb/>co che il pe&longs;o C &longs;i mouerà dalla <lb/>po&longs;&longs;anza in B. </s> <s>Faccia&longs;i AB ad <lb/>AE, come il pe&longs;o C alla po&longs; <lb/>&longs;anza; &longs;arà &longs;imilmente il punto E <lb/>tra BD, percioche egli è nece&longs;&longs;a-<emph.end type="italics"/> <arrow.to.target n="note184"></arrow.to.target><lb/><emph type="italics"/>rio che AE &longs;ia maggiore di A <lb/>D. & &longs;e il pe&longs;o C fo&longs;&longs;e appicca<emph.end type="italics"/> <arrow.to.target n="note185"></arrow.to.target><lb/> <arrow.to.target n="fig77"></arrow.to.target><lb/><emph type="italics"/>to in E, la po&longs;&longs;anza in B lo &longs;o&longs;tentarebbe. </s> <s>ma po&longs;&longs;anza minore po&longs;ta in B, <lb/>che la data &longs;o&longs;tiene il pe&longs;o C appiccato in D; dunque la data po&longs;&longs;anza in B mo-<emph.end type="italics"/> <arrow.to.target n="note186"></arrow.to.target><lb/><emph type="italics"/>uerà il pe&longs;o C appiccato in D con la leua AB, che ha il &longs;uo &longs;o&longs;tegno A.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note184"></margin.target><emph type="italics"/>Per la otta ua del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note185"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note186"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 1. <emph type="italics"/>corollario del la<emph.end type="italics"/> 2. <emph type="italics"/>di que sto.<emph.end type="italics"/></s> </p> <figure id="fig77"></figure> <p type="main"> <s><emph type="italics"/>Sia da capo la leua AB co'l &longs;o&longs;tegno &longs;uo A; & il pe&longs;o C &longs;ia appiccato in B, & <lb/>&longs;ia la po&longs;&longs;anza in D. & DA habbia proportione maggiore ad AB, che il pe <lb/>&longs;o C alla po&longs;&longs;anza in <lb/>D. </s> <s>Dico che il pe&longs;o C<emph.end type="italics"/> <arrow.to.target n="note187"></arrow.to.target><lb/><emph type="italics"/>&longs;arà mo&longs;&longs;o dalla po&longs;&longs;an­<lb/>za in D. </s> <s>faccia&longs;i come <lb/>il pe&longs;o C a'la po&longs;&longs;anza, <lb/>co&longs;i DA &longs;ia ad AE;<emph.end type="italics"/> <arrow.to.target n="note188"></arrow.to.target><lb/><emph type="italics"/>&longs;arà AE maggiore di<emph.end type="italics"/><lb/> <arrow.to.target n="fig78"></arrow.to.target><lb/><emph type="italics"/>AB; per e&longs;&longs;ere proportione maggiore da DA ad AB, che da DA ad AE.<emph.end type="italics"/> <arrow.to.target n="note189"></arrow.to.target><lb/><emph type="italics"/>Che &longs;e il pe&longs;o C &longs;arà appiccato in E, egli è chiaro, che la po&longs;&longs;anza in D &longs;o&longs;ter­<lb/>rà il pe&longs;o C appiccato in E. </s> <s>Ma po&longs;&longs;anza minore che la data &longs;o&longs;tiene l'i&longs;te&longs;&longs;o pe <lb/>&longs;o C in B; dunque la data po&longs;&longs;anza in D mouerà il pe&longs;o C appiccato in B, con <lb/>la leua AB che hà il &longs;o&longs;tegno &longs;uo A. come bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note187"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note188"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note189"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 1. <emph type="italics"/>corollario del la<emph.end type="italics"/> 3. <emph type="italics"/>di que sto.<emph.end type="italics"/></s> </p> <figure id="fig78"></figure> <p type="head"> <s>PROPOSITIONE XII.</s> </p> <p type="head"> <s>PROBLEMA.</s> </p> <p type="main"> <s>Fare che vna data po&longs;&longs;anza, moua vn pe&longs;o dato con vna data le­<lb/>ua. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A come cento, & la po&longs;&longs;anza che ha da mouere &longs;ia come diece; & &longs;ia <lb/>la data leua BC. </s> <s>Egli è bi&longs;ogno che la po&longs;&longs;anza, che è diece moua il pe&longs;o A, che <lb/>è cento, con la leua BC. </s> <s>Diuida&longs;i BC in D con &longs;i fatta maniera che CD hab <lb/>bia la propor tione mede&longs;ima à DB, che ha cento à diece, cioè diece ad vno; per-<emph.end type="italics"/><lb/> <arrow.to.target n="fig79"></arrow.to.target><lb/> <arrow.to.target n="note190"></arrow.to.target> <emph type="italics"/>cioche &longs;e D &longs;i face&longs;&longs;e &longs;o&longs;tegno, egli è mani&longs;e&longs;to, che la po&longs;&longs;anza in C come diece <lb/>pe&longs;erà egualmente co'l pe&longs;o A appiccato in B, cioè che &longs;o&longs;terrà il pe&longs;o A. </s> <s>Pren<emph.end type="italics"/><lb/> <arrow.to.target n="note191"></arrow.to.target> <emph type="italics"/>da&longs;i tra BD qual &longs;i voglia punto, come E, & faccia&longs;i E il &longs;o&longs;tegno. </s> <s>Hor per­<lb/>cioche maggiore è la proportione di CE ad EB, che di CD à DB; CE haurà <lb/>proportione maggiore ad EB, che il pe&longs;o A alla po&longs;&longs;anza di diece po&longs;ta in C; <lb/>dunque la po&longs;&longs;anza di diece po&longs;ta in C mouer à il pe&longs;o A, che è cento, appiccato<emph.end type="italics"/><lb/> <arrow.to.target n="note192"></arrow.to.target> <emph type="italics"/>in B con la leua BC, che ha il &longs;uo &longs;o&longs;tegno E.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note190"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note191"></margin.target><emph type="italics"/>Per lo lein ma di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note192"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig79"></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e la leua fo&longs;&longs;e BC, & il &longs;o&longs;tegno B. diuida&longs;i CB in D per &longs;i fatta maniera, <lb/>che CB habbia la proportione i&longs;te&longs;&longs;a à BD, che ha cento à diece: & &longs;e il pe&longs;o <lb/>A &longs;arà appic <lb/>cato in D, & <lb/>la po&longs;&longs;anza in <lb/>C, la po&longs;&longs;an-<emph.end type="italics"/><lb/> <arrow.to.target n="note193"></arrow.to.target> <emph type="italics"/>za in C come <lb/>diece &longs;o&longs;terrà <lb/>anco il pe&longs;o <lb/>A appiccato<emph.end type="italics"/><lb/> <arrow.to.target n="fig80"></arrow.to.target><lb/><emph type="italics"/>in D. </s> <s>Prenda&longs;i qual &longs;i uoglia punto tra DB, come E, & ponga&longs;i il pe&longs;o A in<emph.end type="italics"/><lb/> <arrow.to.target n="note194"></arrow.to.target> <emph type="italics"/>E; & per e&longs;&longs;ere proportione maggiore da CB à BE, che da BC à BD; CB <lb/>haurà proportione maggiore à BE, che il pe&longs;o A di cento alla po&longs;&longs;anza di diece. </s> <s><lb/>Dunque la po&longs;&longs;anza d<gap/> diece po&longs;ta in C mouerà il pe&longs;o A di cento appiccato in E<emph.end type="italics"/><lb/> <arrow.to.target n="note195"></arrow.to.target> <emph type="italics"/>con la leua BC, che ha il &longs;o&longs;tegno &longs;uo B. che bi&longs;ognaua menar ad effetto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note193"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note194"></margin.target><emph type="italics"/>Per la ottaua del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note195"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig80"></figure> <p type="main"> <s><emph type="italics"/>Ma ciò non &longs;i puote mandar' ad e&longs;ecutione con la leua BC, che habbia il &longs;o&longs;tegno &longs;uo <lb/>in B, & il pe&longs;o A di cento &longs;ia appiccato in C. </s> <s>Percioche ponga&longs;i la po&longs;&longs;anza <lb/>&longs;o&longs;tenente il pe&longs;o A comunque &longs;i &longs;ia tra BC, come in D; &longs;empre la po&longs;&longs;anza <lb/>&longs;arà maggiore del pe&longs;o A. </s> <s>Per laqual co&longs;a egli è me&longs;tieri che &longs;empre la data po&longs;-<emph.end type="italics"/> <pb pagenum="53"/><emph type="italics"/>&longs;anza &longs;ia maggiore del pe&longs;o A. </s> <s>Sia dunque la po&longs;&longs;anza data, come cento cin-<emph.end type="italics"/> <arrow.to.target n="note196"></arrow.to.target><lb/><emph type="italics"/>quanta. </s> <s>Diuida&longs;i BC in D &longs;i fattamente che CB &longs;ia à BD come cento cin­<lb/>quanta à cento, cioè tre à due: & &longs;e la po&longs;&longs;anza &longs;arà po&longs;ta in D, egli è chiaro, <lb/>che la po&longs;&longs;anza in D &longs;o&longs;ter­<lb/>rà il pe&longs;o A appiccato i<gap/> C.<emph.end type="italics"/> <arrow.to.target n="note197"></arrow.to.target><lb/><emph type="italics"/>& co&longs;i prenda&longs;i tra DC <lb/>qual &longs;i voglia punto, come <lb/>E, & ponga&longs;i la po&longs;&longs;aaza <lb/>mouente in E, & per e&longs;&longs;ere <lb/>proportion maggiore da EB <lb/>à BC, che da DB à BC; <lb/>haurà EB proportione mag <lb/>giore à BC, che il pe&longs;o A<emph.end type="italics"/> <arrow.to.target n="note198"></arrow.to.target><lb/> <arrow.to.target n="fig81"></arrow.to.target><lb/><emph type="italics"/>alla po&longs;&longs;anza in E. </s> <s>Dunque la po&longs;&longs;anza di cento cinquanta po&longs;ta in E mouerà il <lb/>pe&longs;o A di cento appiccato in C con la leua BC che hà il &longs;o&longs;tegno B. come bi-<emph.end type="italics"/> <arrow.to.target n="note199"></arrow.to.target><lb/><emph type="italics"/>&longs;ognaua oprare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note196"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 2. <emph type="italics"/>corollario della<emph.end type="italics"/> 3. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note197"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note198"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>del quinto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note199"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig81"></figure> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Di qui è manife&longs;to, &longs;e la data po&longs;&longs;anza &longs;arà maggiore del dato <lb/>pe&longs;o, que&longs;to poter&longs;i fare, ouero &longs;tando in maniera la leua, <lb/>che il &longs;o&longs;tegno &longs;uo &longs;ia fra il pe&longs;o, & la po&longs;&longs;anza; ouero che el­<lb/>la habbia il pe&longs;o fra il &longs;o&longs;tegno, & la po&longs;&longs;anza; ouero alla fine <lb/>e&longs;&longs;endo po&longs;ta la po&longs;&longs;anza fra il pe&longs;o, & il &longs;o&longs;tegno. </s> </p> <p type="main"> <s>Ma &longs;e la data po&longs;&longs;anza &longs;arà minore, ouero eguale al dato pe&longs;o, <lb/>egli è parimente chiaro, che il mede&longs;imo &longs;i puote mandare ad <lb/>e&longs;ecutione &longs;olamente &longs;tando la leua in maniera, che il &longs;o&longs;te­<lb/>gno &longs;uo &longs;ia tra il pe&longs;o, & la po&longs;&longs;anza; ouero che ella habbia il <lb/>pe&longs;o fra il &longs;o&longs;tegno, & la po&longs;&longs;anza. </s> </p> <p type="head"> <s>PROPOSITIONE XIII.</s> </p> <p type="head"> <s>PROBLEMA.</s> </p> <p type="main"> <s>Dati quanti &longs;i <expan abbr="vogliã">vogliam</expan> pe&longs;i appiccati douunque &longs;i &longs;iano nella leua <lb/>il cui &longs;o&longs;tegno parimente &longs;ia dato, ritrouare vna po&longs;&longs;anza la <lb/>quale &longs;o&longs;tenga i dati pe&longs;i in vn punto dato. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Siano i dati pe&longs;i ABC nella leua DE, & il &longs;o&longs;tegno &longs;uo F, douunque ne' pun­<lb/>ti DGH &longs;iano appiccati, & habbia&longs;i à collocare la po&longs;&longs;anza nel punto E. </s> <s>egli <lb/>è me&longs;tieri trouare la po&longs;&longs;anza, laquale &longs;o&longs;tenga in E i dati pe&longs;i ABC con la le <lb/>ua DE. </s> <s>diuida&longs;i DG in K &longs;i fattamente, che DK &longs;ia à KG come il pe­<lb/>&longs;o B al pe&longs;o A; dapoi diuida&longs;i KH in L &longs;i fattamente, che KL &longs;ia ad LH <lb/>come il pe&longs;o C à i pe&longs;i BA, & come FE ad FL, co&longs;i &longs;accian&longs;i i pe&longs;i ABC<emph.end type="italics"/><lb/> <arrow.to.target n="fig82"></arrow.to.target><lb/><emph type="italics"/>tutti in&longs;ieme alla po&longs;&longs;anza, laquale ponga&longs;i in E. dico, che la po&longs;&longs;anza in E. &longs;o-<emph.end type="italics"/><lb/> <arrow.to.target n="note200"></arrow.to.target> <emph type="italics"/>&longs;tenterà i dati pe&longs;i ABC appiccati in DGH con la leua DE che ha il &longs;o&longs;te-<emph.end type="italics"/><lb/> <arrow.to.target n="note201"></arrow.to.target> <emph type="italics"/>gno &longs;uo F. </s> <s>Hor percioche &longs;e i pe&longs;i ABC &longs;o&longs;&longs;ero appiccati in&longs;ieme in L, la po&longs; <lb/>&longs;anza in E &longs;o&longs;terrebbe i dati pe&longs;i appiccati in L; ma i pe&longs;i ABC pe&longs;ano tan­<lb/>to in L, quanto &longs;e C in H, & BA in&longs;ieme &longs;o&longs;&longs;ero appiccati in K; & AB<emph.end type="italics"/><lb/> <arrow.to.target n="note202"></arrow.to.target> <emph type="italics"/>nel K tanto pe&longs;ano, quanto &longs;e A in D, & B in G fo&longs;&longs;ero appiccati; dun­<lb/>què la po&longs;&longs;anza in E &longs;o&longs;tenterà i dati pe&longs;i ABC appiccati in DGH con la <lb/>leua DE che ha il &longs;o&longs;tegno F. </s> <s>Che &longs;e la po&longs;&longs;anza haue&longs;&longs;e ad e&longs;&longs;ere po&longs;ta in qual <lb/>&longs;i voglia altro punto dalla leua DE fuor che in F, come in K; faccia&longs;i come <lb/>FK ad FL, co&longs;ii pe&longs;i ABC &longs;iano alla po&longs;&longs;anza: &longs;imilmente dimo&longs;treremo, <lb/>che la po&longs;&longs;anza in K &longs;o&longs;terrà i pe&longs;i ABC ne' punti DGH appiccati. </s> <s>come <lb/>bi&longs;ognaua fare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note200"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note201"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo della bilancia.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note202"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig82"></figure> <p type="main"> <s><emph type="italics"/>Da que&longs;ta, & dalla quinta di que&longs;to, &longs;e i pe&longs;i ABC &longs;aranno po&longs;ti in qual &longs;i voglia <lb/>modo nella leua DE, & che bi&longs;ogni ritrouare la po&longs;&longs;anza, la quale debba &longs;o&longs;te­<lb/>nere in E i dati pe&longs;i &longs;iano tirate da i centri delle grauezze de i pe&longs;i le linee AB <lb/>C à piombo de gli orizonti, lequali taglino la leua DE ne' punti DGH; &<emph.end type="italics"/> <pb pagenum="54"/><emph type="italics"/>&longs;i operino le altre co&longs;e nell'i&longs;te&longs;&longs;o modo: egli è manife&longs;to, che la po&longs;&longs;anza in E,<emph.end type="italics"/><lb/> <arrow.to.target n="fig83"></arrow.to.target><lb/><emph type="italics"/>ouero in K &longs;o&longs;tenterà i dati pe&longs;i, percioche egli è l'i&longs;te&longs;&longs;o come &longs;e i pe&longs;i fo&longs;&longs;ero <lb/>appiccati in DGH.<emph.end type="italics"/></s> </p> <figure id="fig83"></figure> <p type="head"> <s>PROPOSITIONE XIIII.</s> </p> <p type="head"> <s>PROBLEMA.</s> </p> <p type="main"> <s>Fare che vna data po&longs;&longs;anza moua quanti pe&longs;i &longs;i vogliano, po&longs;ti <lb/>douunque, & in qualunque modo &longs;i &longs;ia in vna data leua. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la data leua DE, & &longs;iano i dati pe&longs;i, come è po&longs;to nel precedente corollario, & <lb/>&longs;ia A come cento, B come cinquanta, & C come trenta; & la data po&longs;&longs;an­<lb/>za &longs;ia come trenta. </s> <s>&longs;iano po&longs;te le co&longs;e mede&longs;ime, & ritroui&longs;i il punto L; dapoi<emph.end type="italics"/><lb/> <arrow.to.target n="fig84"></arrow.to.target><lb/><emph type="italics"/>diuida&longs;i LE in F, &longs;i &longs;attamente che FE ad FL &longs;ia come cento ottanta à <lb/>trenta, cioè &longs;ei ad vno, & &longs;e F &longs;i face&longs;&longs;e &longs;o&longs;tegno, la po&longs;&longs;anza come trenta<emph.end type="italics"/> <pb/> <arrow.to.target n="note203"></arrow.to.target> <emph type="italics"/>in E &longs;o&longs;terrebbe i pe&longs;i ABC. </s> <s>pigli&longs;i dunque tra LF qualunque punto come <lb/>M, & faccia&longs;i M il &longs;o&longs;tegno: egli è manife&longs;to, che la po&longs;&longs;anza po&longs;ta in E co-<emph.end type="italics"/><lb/> <arrow.to.target n="fig85"></arrow.to.target><lb/> <arrow.to.target n="note204"></arrow.to.target> <emph type="italics"/>me trenta mouerài pe&longs;i ABC come cento ottanta con la leua DE. che bi&longs;o­<lb/>gnaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note203"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 13. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note204"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig84"></figure> <figure id="fig85"></figure> <p type="main"> <s><emph type="italics"/>Ma ciò non potremo già vniuer&longs;almente menare ad ef&longs;etto, &longs;e il &longs;o&longs;tegno &longs;o&longs;&longs;e nelle <lb/>&longs;tremità della leua, come in D; peroche la proportione di DE à DL, cioè la <lb/>proportione de' pe&longs;i ABC alla po&longs;&longs;anza, laquale ha da &longs;o&longs;tenere i pe&longs;i &longs;empre <lb/>è data. </s> <s>Laqual co&longs;a molto meno anco &longs;i potrebbe fare, &longs;e la po&longs;&longs;anza &longs;i haue&longs;&longs;e <lb/>à porre tra DL.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE XV.</s> </p> <p type="head"> <s>PROBLEMA.</s> </p> <p type="main"> <s>Ma percioche mentre i pe&longs;i &longs;i mouono con la leua, ha la leua an­<lb/>cora grauezza, della quale infin qui non &longs;i è fatto mentione <lb/>alcuna: però dimo&longs;triamo primieramente in che modo &longs;i tro <lb/>ui la po&longs;&longs;anza, la quale &longs;o&longs;tenga nel dato punto la leua data, il <lb/>cui &longs;o&longs;tegno &longs;ia parimente dato. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua data AB, il cui &longs;o&longs;tegno C &longs;ia dato: & &longs;ia il punto D nelquale &longs;i hab <lb/>bia à collocare la po&longs;&longs;anza, che debba &longs;o&longs;tentare la leua AB, &longs;i &longs;attamente che <lb/>re&longs;ti immobile. </s> <s>&longs;ia dal punto C tirata la linea CE à piombo dell'orizonte la <lb/>quale diuida la leua AB in due parti AE EF; & della parte AE &longs;ia il <lb/>centro G della grauezza, & della parte EF il centro del'a grauezza &longs;ia H, <lb/>& dai punti GH &longs;iano tirate le linee GK HL à piombo de gli orizonti, le<emph.end type="italics"/> <pb pagenum="55"/><emph type="italics"/>quali taglino la linea AF ne' punti KL. </s> <s>Hor percioche la leua AB è diui­<lb/>&longs;a dalla linea CE in due parti, cioè AE EF; però la leua AB, niente altro <lb/>&longs;arà, che due pe&longs;i AE EF nella leua, ouero bilancia AF po&longs;ti; il cui appicca <lb/>mento, ouero &longs;o&longs;tegno è C. </s> <s>Per laqual co&longs;a i pe&longs;i AE EF &longs;aranno co&longs;i po&longs;ti,<emph.end type="italics"/><lb/> <arrow.to.target n="fig86"></arrow.to.target><lb/><emph type="italics"/>come &longs;e fo&longs;&longs;ero appiccati in KL. </s> <s>Diuida&longs;i dunque KL in M, &longs;i fattamente, <lb/>che KM &longs;ia ad ML come la grauezza della parte EF alla grauezza della <lb/>parte AE; & come CA à CM, co&longs;i &longs;accia&longs;i la grauezza di tutta la leua <lb/>AB alla po&longs;&longs;anza, laquale &longs;e in D &longs;arà collocata (pur che DA &longs;ia à piombo<emph.end type="italics"/> <arrow.to.target n="note205"></arrow.to.target><lb/><emph type="italics"/>dell'orizonte) pe&longs;erà egualmente con la leua; cioè &longs;o&longs;terrà la leua AB premendo <lb/>in giù, che bi&longs;ognaua trouare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note205"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 13. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig86"></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e la po&longs;&longs;anza &longs;i haue&longs;&longs;e à porre nel punto B. </s> <s>Faccia&longs;i come CF à CM, <lb/>co&longs;i il pe&longs;o AB alla po&longs;&longs;anza. </s> <s>Con &longs;imile modo prouera&longs;&longs;i che la po&longs;&longs;anza in <lb/>B &longs;o&longs;terrà la leua AB. & l'i&longs;te&longs;&longs;o d mo&longs;trera&longs;&longs;i in qualunque altro &longs;ito s'haue&longs;­<lb/>&longs;e à porre la po&longs;&longs;anza, (fuor che in E) come in N. peroche faccia&longs;i CO à <lb/>CM come AB alla po&longs;&longs;anza, laquale &longs;e &longs;i porrà in N &longs;o&longs;tenterà la leua AB.<emph.end type="italics"/></s> </p> <p type="main"> <s>Ma aggiunga &longs;i il pe&longs;o appiccato, ouero po&longs;to nella leua; come, <lb/>po&longs;te le co&longs;e i&longs;te&longs;fe, fia il pe&longs;o P appiccato in A; & la po&longs;­<lb/>&longs;anza s'habbia à porre in B, &longs;i fattamente che &longs;o&longs;tenghi la le <lb/>ua AB in&longs;ieme col pe&longs;o P. </s> </p> <p type="main"> <s><emph type="italics"/>Diuida&longs;i AM in Q, &longs;i &longs;attamente, che AQ &longs;ia à QM, come la grauezza<emph.end type="italics"/> <arrow.to.target n="note206"></arrow.to.target><lb/><emph type="italics"/>della leua AB alla grauezza del pe&longs;o P; dapoi come CF à CQ, co&longs;i fac-<emph.end type="italics"/> <arrow.to.target n="note207"></arrow.to.target><lb/><emph type="italics"/>cia&longs;i la grauezza AB, & P in&longs;ieme alla po&longs;&longs;anza, la quale ponga&longs;i in B: e<gap/>li <lb/>è manife&longs;to, che la po&longs;&longs;anza in B &longs;o&longs;terrà la leua AB in&longs;ieme co'l pe&longs;o P. </s> <s>Che <lb/>&longs;e fo&longs;&longs;e CA à CM, come AB à P; &longs;arebbe il punto C il loro centro della <lb/>grauezza, & perciò la leua AB in&longs;ieme co'l pe&longs;o P &longs;enza la po&longs;&longs;anza po&longs;ta in <pb/>B &longs;tarà &longs;erma. </s> <s>Ma &longs;e il centro della grauezza de' pe&longs;i fo&longs;&longs;e tra CF, come in <lb/>O. </s> <s>Faccia&longs;i come CF à CO, co&longs;i AB & P in&longs;ieme alla po&longs;&longs;anza, laqua­<lb/>le in B &longs;o&longs;tenterà sì la leua AB come il pe&longs;o P.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note206"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 13 <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note207"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/><gap/> Archirnc<gap/> dells col che egual meiue pe&longs;e no.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Similmente mo&longs;trera&longs;&longs;i il mede&longs;imo &longs;e fo&longs;&longs;ero più pe&longs;i nella leua AB douunque, <lb/>& in qual modo &longs;i &longs;ia di&longs;po&longs;ti.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Oltre à ciò da que&longs;te co&longs;e &longs;i puote cono&longs;cere, come nella decimaquarta propo&longs;itione di <lb/>que&longs;to habbiamo in&longs;egnato, in che modo cioè po&longs;&longs;iamo mouere i dati pe&longs;i po&longs;ti do <lb/>uunque &longs;i voglia nella leua, con vna data po&longs;&longs;anza, e con vna data leua, ilche po&longs;­<lb/>&longs;iamo fare nell'i&longs;te&longs;&longs;o modo non &longs;olamente con&longs;iderando la grauezza della leua; ma <lb/>anco gli altri accidenti, iquali &longs;ono &longs;tati di &longs;opra mo&longs;trati &longs;enza la grauezza del­<lb/>la leua; con &longs;imile modo con&longs;iderata la grauezza della leua in&longs;ieme co' pe&longs;i, ouero <lb/>&longs;enza pe&longs;i &longs;i mo&longs;treranno.<emph.end type="italics"/></s> </p> <p type="head"> <s>IL FINE DELLA LEVA.</s> </p> <pb pagenum="56"/> <figure></figure> <p type="head"> <s>DELLA TAGLIA.</s> </p> <figure></figure> <p type="main"> <s>Con l'in&longs;trumento della Taglia &longs;i può mouere il pe <lb/>&longs;o in molti modi: ma percioche in tutti è la ragio­<lb/>ne mede&longs;ima: però affine che la co&longs;a re&longs;ti più chia­<lb/>ra, intenda&longs;i in quello che &longs;i ha da dire, che il pe­<lb/>&longs;o &longs;empre &longs;i habbia da mouere all'insù ad angoli <lb/>retti al piano dell'orizonte in que&longs;to modo. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A ilquale &longs;i habbia ad alzare in sù ad angoli retti al piano dell'orizonte: <lb/>& come &longs;i co&longs;tuma di fare: &longs;ia <lb/>aitaccata di &longs;opra vna taglia, <lb/>che habbia due girelle, gli a&longs;&longs;etti <lb/>de<gap/>lequali &longs;iano in BC: & &longs;ia <lb/>anche legata vn'altra taglia al <lb/>pe&longs;o, laquale &longs;imilmente habbia <lb/>due girelle, gli a&longs;&longs;etti delle qua­<lb/>li &longs;<gap/>no in DE: & per tutte <lb/>le girelle d'ambedue le taglie &longs;ia <lb/>condo<gap/>a intorno la corda, la­<lb/>quale in vno de i capi, come in <lb/>F deue e&longs;&longs;ere legata. </s> <s>Ponga&longs;i <lb/>ancorala po&longs;&longs;anza che moue in <lb/>G, laquale mentre di&longs;cende, il <lb/>pe&longs;o A per lo contrario &longs;arà le­<lb/>uato in &longs;u&longs;o, &longs;i come afferma Pa <lb/>po nell'ottauo libro delle rac­<lb/>colte matematiche, & Vitruuio <lb/>nel decimo dell'architettura, & <lb/>altri.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s>Hor in che modo que&longs;to <lb/>in&longs;trumento della ta­<lb/>glia &longs;i riduca alla leua, <lb/>& perche vn pe&longs;o gran­<lb/>de &longs;i moua da piccola <lb/>forza, & in qual modo, <lb/>& in quanto tempo; & <lb/>perche la corda debba <lb/>e&longs;&longs;ere legata da vn ca­<lb/>po: & quale debba e&longs;­<lb/>&longs;ere l'officio della ta­<lb/>glia, che è po&longs;ta di &longs;ot­<lb/>to, & quale di quella, <lb/>che &longs;tà di &longs;opra, & in <lb/>che modo &longs;i po&longs;&longs;a tro­<lb/>uare ogni proportio­<lb/>ne data ne i numeri tra la po&longs;&longs;anza, & il pe&longs;o, diciamo. </s> </p> <pb pagenum="57"/> <p type="head"> <s>LEMMA.</s> </p> <p type="main"> <s>Siano due linee rette AB CD egualmente di&longs;tanti, lequali <lb/>tocchino il cerchio ACE ne' punti AC, il centro delqual <lb/>cerchio &longs;ia F, & &longs;i congiunghino FA & FC. </s> <s>dico chela <lb/>linea AFC è retta. </s> </p> <p type="main"> <s><emph type="italics"/>Tiri&longs;i la linea FE egualmente di&longs;tante dal-<emph.end type="italics"/> <arrow.to.target n="note208"></arrow.to.target><lb/><emph type="italics"/>le linee AB CD. </s> <s>Et percioche AB <lb/>& FE &longs;ono egualmente di&longs;tanti, &<emph.end type="italics"/> <arrow.to.target n="note209"></arrow.to.target><lb/><emph type="italics"/>l'angolo BAF èretto: &longs;arà anco A <lb/>FE retto, & all'ifte&longs;&longs;o modo CFE &longs;a<emph.end type="italics"/> <arrow.to.target n="note210"></arrow.to.target><lb/><emph type="italics"/>rà retto: adunque la linea AFC èret­<lb/>ta, ilche s'hauea à dimo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note208"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del terzo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note209"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note210"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 14. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>PROPOSITIONE I.</s> </p> <p type="main"> <s>Se la corda &longs;i condurrà intorno alla girella della taglia, che &longs;ia <lb/>attaccata di &longs;opra, & che vno delli &longs;uoi capi &longs;i leghi al pe&longs;o, & <lb/>l'altro tratanto &longs;ia pre&longs;o dalla po&longs;sanza, che &longs;o&longs;tiene il detto <lb/>pe&longs;o; la po&longs;sanza &longs;arà eguale al pe&longs;o. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A alquale venga legata la corda à B: & la taglia, che habbia la girella <lb/>CEF il cui centro D appicchi&longs;i di &longs;opra: & &longs;ia parimente D il centro dell'a&longs; <lb/>&longs;etto, & d'intorno alla girella volga&longs;i la corda BCEFG: & &longs;ia in G la po&longs;­<lb/>&longs;anza, che &longs;o&longs;tiene il pe&longs;o A. </s> <s>Dico la po&longs;&longs;anza po&longs;tain G e&longs;&longs;ere eguale al pe­<lb/>&longs;o A. </s> <s>&longs;ia FG <lb/>egualmente di­<lb/>&longs;tante da CB. </s> <s><lb/>Percioche <expan abbr="dū-que">dun­<lb/>que</expan> il pe&longs;o A <lb/>&longs;ta fermo, &longs;a-<emph.end type="italics"/><lb/> <arrow.to.target n="note211"></arrow.to.target> <emph type="italics"/>rà CB à piom <lb/>bo del piano <lb/>dell'orizonte.<emph.end type="italics"/><lb/> <arrow.to.target n="note212"></arrow.to.target> <emph type="italics"/>onde FG &longs;a­<lb/>rà al piano <lb/>i&longs;te&longs;&longs;o à piom­<lb/>bo. </s> <s>Siano i <lb/>punti CF nel­<lb/>la girella, da <lb/>quali le corde <lb/>CB FG &longs;cen <lb/>dano nel pia­<lb/>no dell'orizon <lb/>te ad angoli <lb/>retti, tocche­<lb/>ranno le dette <lb/>corde BC FG <lb/>la girella CE<emph.end type="italics"/><lb/> <arrow.to.target n="fig87"></arrow.to.target><lb/><emph type="italics"/>F ne'punti CF peroche non po&longs;&longs;ono &longs;egare la girella. </s> <s>Siano congiunte le li-<emph.end type="italics"/><lb/> <arrow.to.target n="note213"></arrow.to.target> <emph type="italics"/>nee DC DF. &longs;arà retta la linea CF & &longs;aranno anche retti gli ang oli DCB <lb/>DFG. </s> <s>Ma percioche BC &longs;ta à piombo sì all'orizonte, come ad e&longs;&longs;a CF &longs;arà<emph.end type="italics"/><lb/> <arrow.to.target n="note214"></arrow.to.target> <emph type="italics"/>la detta CF egualmente di&longs;tante dall'orizonte. </s> <s>& concio&longs;ia che il pe&longs;o &longs;ia attac­<lb/>cato in CB & la po&longs;&longs;anza &longs;ia in G ch'è il mede&longs;imo, come &longs;e ella fo&longs;&longs;e in F:<emph.end type="italics"/><lb/> <arrow.to.target n="note215"></arrow.to.target> <emph type="italics"/>&longs;arà CF tanto quanto vna bilancia, ouero vna leua, il cui centro, ouero &longs;o&longs;tegno <lb/>&longs;arà D, imperoche la girella è &longs;o&longs;tenuta nell'a&longs;&longs;etto, & il punto D per e&longs;&longs;ere <lb/>centro dell'a&longs;&longs;etto, & della girellarimane immobile, &longs;eben l'vno, & l'altro &longs;ivol <lb/>gono intorno. </s> <s>Per laqual co&longs;a e&longs;&longs;endo la di&longs;tanza DC eguale alla di&longs;tanza DF, <lb/>& la po&longs;&longs;anza che è in F contrape&longs;i egualmente al pe&longs;o A attaccato in C &longs;o­<lb/>&longs;tenendo il pe&longs;o in modo, che non cala al ba&longs;&longs;o, &longs;arà la po&longs;&longs;anza a&longs;&longs;egnata in F oue <lb/>ro in G che è tutt'vno, eguale al pe&longs;o A: percioche po&longs;ta in G fal'i&longs;te&longs;&longs;o effet <lb/>to che &longs;e nel mede&longs;imo G &longs;o&longs;&longs;e appiccato vn'altro pe&longs;o eguale al pe&longs;o A, liquali <lb/>pe&longs;i attaccati in CF contrape&longs;erann<gap/> egualmente. </s> <s>Oltre à ciò non facendo&longs;i moto<emph.end type="italics"/> <pb pagenum="58"/><emph type="italics"/>in niuna delle parti, &longs;arà l'i&longs;te&longs;&longs;o e&longs;&longs;endo circondata in que&longs;to modo la girella intor­<lb/>no con vna corda &longs;ola BC e FG come &longs;e &longs;u&longs;&longs;ero due corde BC FG legate <lb/>alla leua, ouero alla bilancia CF.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note211"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo della bilancia.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note212"></margin.target><emph type="italics"/>Per la <gap/>ta <gap/>a dell'vndecimo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note213"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>del terzo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note214"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 28. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note215"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>del<emph.end type="italics"/> 1. <emph type="italics"/>d'Archimede delle co&longs;e che pe&longs;ano egual mem<gap/>:.<emph.end type="italics"/></s> </p> <figure id="fig87"></figure> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Da que&longs;to può e&longs;&longs;ere manife&longs;to, che il mede&longs;imo pe&longs;o dalla i&longs;te&longs;­<lb/>&longs;a po&longs;&longs;anza puote e&longs;&longs;ere tuttauia &longs;o&longs;tenuto &longs;enza anche alcu­<lb/>no aiuto di que&longs;ta taglia. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche &longs;ia il pe&longs;o H eguale al pe&longs;o A à cui &longs;ia legata la corda KL & &longs;iala <lb/>po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o H in L. </s> <s>Hor concio&longs;ia che volendo &longs;o&longs;tenere al­<lb/>cun pe&longs;o &longs;enza aiuto veruno vi bi&longs;ogni tanta forza, quanta &longs;ia eguale al pe&longs;o; la <lb/>po&longs;&longs;anza che è in L &longs;arà eguale al <lb/>pe&longs;o H, ma il pe&longs;o H è po&longs;to <lb/>eguale al pe&longs;o A, alquale è anco <lb/>eguale la po&longs;&longs;anza G. </s> <s>&longs;arà dun­<lb/>que la po&longs;&longs;anza in G eguale alla <lb/>po&longs;&longs;anza in L che è l'i&longs;te&longs;&longs;o, come &longs;e <lb/>la i&longs;te&longs;&longs;a po&longs;&longs;anza &longs;o&longs;tene&longs;&longs;e il pe&longs;o <lb/>mede&longs;imo. </s> <s>Oltre à ciò &longs;e le po&longs;&longs;an <lb/>ze, lequali &longs;ono in G & in L &longs;o&longs; <lb/>&longs;ero eguali fra loro, & poi &longs;epara­<lb/>tamente dai pe&longs;i minori, è co&longs;a chia <lb/>ra, che le dette po&longs;&longs;anze non &longs;areb <lb/>bono &longs;ufficienti à &longs;o&longs;tenere quei pe&longs;i <lb/>che &longs;e que&longs;te po&longs;&longs;anze &longs;aranno mag <lb/>giori, egli è manife&longs;to, che e&longs;&longs;e mo­<lb/>ueranno i pe&longs;i. </s> <s>& co&longs;i la po&longs;&longs;anza <lb/>in L col pe&longs;o H venirà ad e&longs;&longs;e­<lb/>re nella proportione mede&longs;ima, co­<lb/>me la po&longs;&longs;anza in G col pe&longs;o A.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma perche nella dimo&longs;tratione è &longs;tato <lb/>pre&longs;uppo&longs;to che l'a&longs;&longs;etto &longs;i volga in <lb/>torno, ilquale il più delle volte &longs;tà <lb/>immobile, però &longs;tando anche immobile il detto a&longs;&longs;etto dimo&longs;tri&longs;i li&longs;te&longs;&longs;o.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia la girella della taglia CEF, il cui centro &longs;ia D, & &longs;ia l'a&longs;&longs;etto GHK, il cen­<lb/>tro delquale &longs;ia mede&longs;imamenie D: Tiri&longs;i il diametro CGDKF egualmente <lb/>di&longs;tante dall'orizonte. </s> <s>et percioche <expan abbr="mē">men</expan> <lb/>tre la girella &longs;i volge, la circonferenza <lb/>del cerchio CEF &longs;empre va egual­<lb/>mente di&longs;tante alla circonferenza del­<lb/>l'a&longs;&longs;etto GHK: percioche ella &longs;i <lb/>volge intorno à l'a&longs;&longs;etto, & le circonfe <lb/>renze de' cerchi egualmente di&longs;tanti <lb/>hanno il centro mede&longs;imo, &longs;arà il pun­<lb/>to D &longs;empre centro & della girella, <lb/>& dell'a&longs;&longs;etto. </s> <s>Per laqual co&longs;a e&longs;&longs;en­<lb/>do DC eguale à DF & DG ad <lb/>e&longs;&longs;o DK, &longs;arà GC ad e&longs;&longs;o KF egua <lb/>le. </s> <s>Se dunque nella leua, ouero bilan­<lb/>cia CF &longs;i attaccheranno pe&longs;i eguali, <lb/>contrape&longs;eranno egualmente, peroche <lb/>la di&longs;tanza CG è eguale alla di&longs;tan­<lb/>za KF, & l'a&longs;&longs;etto GHK immobi <lb/>le &longs;erue per centro, ouero per &longs;o&longs;tegno. </s> <s>Stando dunque immobile l'a&longs;&longs;etto, &longs;e la <lb/>po&longs;&longs;anza &longs;i metterà in F che &longs;o&longs;tenga il pe&longs;o appiccato in C, &longs;arà la po&longs;&longs;anza <lb/>in F ad e&longs;&longs;o pe&longs;o eguale, ilche era da mo&longs;trare.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Et concio&longs;ia che del tutto &longs;ia il mede&longs;imo, che l'a&longs;&longs;etto ouero &longs;i volga intorno, ò non <lb/>&longs;i volga: però &longs;ialecito nelle co&longs;e, che &longs;i hanno à dire, prendere in loco dello a&longs;&longs;etto <lb/>il centro &longs;olamente.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE II.</s> </p> <p type="main"> <s>Se la corda &longs;i condurrà intorno alla girella della taglia, che &longs;ia <lb/>legata al pe&longs;o, legando l'vn de' capi &longs;uoi in qualche loco, & <lb/>l'altro &longs;ia pre&longs;o dalla po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o, &longs;arà la po&longs;­<lb/>&longs;anza la metà meno del pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A. </s> <s>&longs;ia BCD la girella della taglia legata al pe&longs;o, il cui centro &longs;ia E, <lb/>&longs;ia dapoi inuolta d'intorno la girella la corda FBCDG, & legata in F, & &longs;ia <lb/>la po&longs;&longs;anza in G che &longs;o&longs;tiene il pe&longs;o A. </s> <s>Dico che la po&longs;&longs;anza in G è la metà <lb/>meno del pe&longs;o A. </s> <s>Siano le corde FB GD perpendicolari all' orizonte del pun<emph.end type="italics"/><lb/> <arrow.to.target n="note216"></arrow.to.target> <emph type="italics"/>to E, lequali &longs;aranno fra loro egualmente di&longs;tanti: & tocchino le dette corde <lb/>FBGD, il cerchio BCD ne i punti BD: congiunga&longs;i la linea BD ella pa&longs;-<emph.end type="italics"/> <pb pagenum="59"/><emph type="italics"/>&longs;erà per E centro, & &longs;arà egualmente di&longs;tante dall'orìzonte di e&longs;&longs;o centro, &<emph.end type="italics"/> <arrow.to.target n="note217"></arrow.to.target><lb/><emph type="italics"/>concio&longs;ia che la G po&longs;&longs;anza debba &longs;o&longs;tenere il pe&longs;o A con la taglia; bi&longs;ogna, <lb/>che la corda &longs;ia legata dal'vno de' capi, come in<emph.end type="italics"/> F, <emph type="italics"/>&longs;i fattamente, che F fac­<lb/>cia re&longs;i&longs;tenza egualmente almeno alla po&longs;&longs;anza, ch'è in G, altramente e&longs;&longs;a po&longs;&longs;an <lb/>zain G non potrebbe à modo alcuno &longs;o&longs;tenere il pe&longs;o. </s> <s>Et perche la po&longs;&longs;anza<emph.end type="italics"/><lb/> <arrow.to.target n="fig88"></arrow.to.target><lb/><emph type="italics"/>&longs;o&longs;tiene la girella mediante la corda, & la girella &longs;o&longs;tiene la parte re&longs;tante della <lb/>taglia mediante l'a&longs;&longs;etto, allaqual taglia il pe&longs;o è appiccato, pe&longs;erà que&longs;ta parte del­<lb/>la taglia nell'a&longs;&longs;etto, cioè nel centro E: onde il pe&longs;o A pe&longs;erà &longs;imilmente nel me <lb/>de&longs;imo centro E, come &longs;e egli fo&longs;&longs;e appiccato in E. </s> <s>Po&longs;ta dunque la po&longs;&longs;auza <lb/>che stà in G doue è D (perche egli è totalmente il mede&longs;imo) &longs;arà BD come <lb/>vna lèua, il cui &longs;o&longs;tegno &longs;arà B, & il pe&longs;o attaccato in E, & la po&longs;&longs;anzain D: <lb/>& e&longs;&longs;endo la corda FB immobile, conueneuolmente il B puote &longs;eruire per &longs;o­<lb/>&longs;tegno. </s> <s>Ma ciò più chiaramente apparerà dapoi. </s> <s>Hora percioche la po&longs;&longs;anza al<emph.end type="italics"/> <arrow.to.target n="note218"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o ha la proportione mede&longs;ima, che hà BE à BD, & BE in proportione <lb/>è la metà manco di BD: dunque la po&longs;&longs;anza che è in G &longs;arà la metà meno del <lb/>pe&longs;o A. </s> <s>Che bi&longs;ognaua dimo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note216"></margin.target><emph type="italics"/>Per la &longs;esta dell'vndecimo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note217"></margin.target><emph type="italics"/>Per la procedense.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note218"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/><gap/> questo nella leua.<emph.end type="italics"/></s> </p> <figure id="fig88"></figure> <p type="main"> <s><emph type="italics"/>Que&longs;to dunque &longs;tà nell'i&longs;te&longs;&longs;o modo con vna corda &longs;ola FBCDG condotta intor­<lb/>no alla girella, come &longs;e &longs;o&longs;&longs;ero due corde BF GD legate alla leua BD, il cui<emph.end type="italics"/><lb/> <arrow.to.target n="fig89"></arrow.to.target><lb/><emph type="italics"/>&longs;o&longs;tegno &longs;arà B, & il pe&longs;o fo&longs;&longs;e attaccato in E & la po&longs;&longs;anza, che lo &longs;o&longs;tiene <lb/>&longs;o&longs;&longs;e in D, ouero in G che è l'iste&longs;&longs;o.<emph.end type="italics"/></s> </p> <figure id="fig89"></figure> <p type="head"> <s>COROLLARIO I.</s> </p> <p type="main"> <s>Da que&longs;to dunque è manife&longs;to, che il pe&longs;o è &longs;o&longs;tenuto à que&longs;to <lb/>modo da po&longs;&longs;anza minore in proportlone della metà meno, <lb/>di quel che &longs;arebbe &longs;enza aiuto veruno di cotale taglia. </s> </p> <pb pagenum="60"/> <p type="main"> <s><emph type="italics"/>Come &longs;ia il pe&longs;o H eguale al <lb/>pe&longs;o A, alquale &longs;ia lega­<lb/>tala corda KL, & la po&longs;­<lb/>&longs;anza, che è in L &longs;o&longs;ten­<lb/>gail pe&longs;o H, &longs;arà la po&longs;­<lb/>&longs;anza in L &longs;eparatamente <lb/>eguale al pe&longs;o H, & al <lb/>pe&longs;o A; ma la po&longs;&longs;an­<lb/>za, che è in G in propor­<lb/>tione è la metà manco del <lb/>pe&longs;o A. </s> <s>Per laqual co&longs;a <lb/>la po&longs;&longs;anzache è in G &longs;a­<lb/>rà la metà meno in propor­<lb/>tione della po&longs;&longs;anza, che è <lb/>in L, & in que&longs;to modo <lb/>ne gli altri tutti di que&longs;ta <lb/>maniera &longs;i potrà ritrouare <lb/>la proportione.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>COROLLARIO II.</s> </p> <p type="main"> <s>Egli è manife&longs;to ancora, &longs;e &longs;aranno due po&longs;&longs;anze l'vna in G & <lb/>l'altra in F, lequali &longs;o&longs;tengano il pe&longs;o A, che l'vna, & l'al­<lb/>tra in&longs;ieme &longs;aranno eguali al pe&longs;o A, & cia&longs;cheduna di loro <lb/>&longs;o&longs;terrà la metà del pe&longs;o A. </s> </p> <p type="main"> <s><emph type="italics"/>Et que&longs;to è mani&longs;e&longs;to dal terzo & dal quarto corollario del &longs;econdo di que&longs;tonel trat. </s> <s><lb/>tato della leua.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO III.</s> </p> <p type="main"> <s>Oltre à ciò que&longs;to parimente &longs;i fa noto, perche cioè la corda <lb/>debba e&longs;&longs;ere legata nell'vno de' capi. </s> </p> <p type="head"> <s>PROPOSITIONE III.</s> </p> <p type="main"> <s>Se à cia&longs;cuna dell'vna, & l'altra girella delle due taglie, l'vna del <lb/>le quali &longs;ia po&longs;ta di &longs;opra, & l'altra di &longs;otto, & que&longs;ta &longs;ia lega­<lb/>ta al pe&longs;o; &longs;arà condotta intorno la corda: legando l'vno de' <lb/>capi in qualche loco, & l'altro &longs;ia tenuto dalla po&longs;&longs;anza, che <lb/>&longs;o&longs;tiene il pe&longs;o, &longs;arà la po&longs;&longs;anza la metà meno del pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A, &longs;ia BCD la girella della <lb/>taglia, che &longs;ia legata al pe&longs;o A, il cui <lb/>centro &longs;ia K, & EFG &longs;ia la girella <lb/>della taglia appiccata di opra, il cui cen­<lb/>tro &longs;ia H, dapoi &longs;ia condotta intorno <lb/>le girelle la corda LBCDMEFGN <lb/>laquale &longs;ia legata in L, & &longs;ia la po&longs;­<lb/>&longs;anza, che &longs;o&longs;tiene il pe&longs;o A in N. </s> <s><lb/>Dico la po&longs;&longs;anza, che &longs;ta in N e&longs;&longs;e­<lb/>re la metà meno del pe&longs;o A. </s> <s>Percio­<lb/>che &longs;e la po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o<emph.end type="italics"/><lb/> <arrow.to.target n="note219"></arrow.to.target> <emph type="italics"/>A fo&longs;&longs;e collocata doue &longs;ta M, &longs;areb­<lb/>be per certo la po&longs;&longs;anza in M la metà<emph.end type="italics"/><lb/> <arrow.to.target n="note220"></arrow.to.target> <emph type="italics"/>meno del pe&longs;o A: & alla po&longs;&longs;anza in <lb/>M è eguale la forza di N, percioche <lb/>egli è come &longs;e la po&longs;&longs;anza in M &longs;o&longs;te­<lb/>ne&longs;&longs;e la metà del pe&longs;o A &longs;enza taglia, <lb/>alquale egualmente contrape&longs;a il pe&longs;o che <lb/>è in N per e&longs;&longs;ere eguale alla metà del <lb/>pe&longs;o A. </s> <s>Per laqual co&longs;ala forza in N <lb/>che è alla metà del pe&longs;o A eguale, &longs;o­<lb/>&longs;tenirà e&longs;&longs;o A. </s> <s>La po&longs;&longs;anza dunque in <lb/>N che &longs;o&longs;tiene il pe&longs;o A, è la metà <lb/>meno di e&longs;&longs;o A. che bi&longs;ognaua mo. <lb/><gap/>rare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note219"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note220"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="61"/> <p type="main"> <s><emph type="italics"/>Ma &longs;e, come nella &longs;econda figura, la cor <lb/>da BCDEFGHKL &longs;arà inuolta <lb/>d'intorno à le girelle, & legata in <lb/>B: & la po&longs;&longs;anza in L &longs;o&longs;tenga il <lb/>pe&longs;o A, &longs;arà &longs;imilmente la po&longs;&longs;an <lb/>za in L la metà meno del pe&longs;o: <lb/>Peroche la girella della taglia di &longs;o­<lb/>pra, & la taglia i&longs;te&longs;&longs;a &longs;ono del tut­<lb/>to inutili: & è il mede&longs;imo, come &longs;e <lb/>la corda fo&longs;&longs;e legata in F, & che <lb/>la po&longs;&longs;anza in L &longs;o&longs;tene&longs;&longs;e il pe&longs;o <lb/>con la &longs;ola taglia legata al pe&longs;o, la <lb/>qual po&longs;&longs;anza è &longs;tata dimo&longs;trata e&longs;­<lb/>&longs;ere la metà meno del pe&longs;o A.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Seguita da que&longs;te co&longs;e, che &longs;e &longs;aranno due po&longs;&longs;anze in BL, am­<lb/>bedue tra loro &longs;aranno eguali. </s> </p> <p type="main"> <s><emph type="italics"/>Perciocheogn'vna di loro da per &longs;e è la metàmeno di e&longs;&longs;o A.<emph.end type="italics"/></s> </p> <pb/> <p type="head"> <s>PROPOSITIONE IIII.</s> </p> <p type="main"> <s>Sia la leua AB, il cui &longs;o&longs;tegno &longs;ia A, laqual leua &longs;ia diui&longs;a in <lb/>due parti eguali in D, & &longs;ia il pe&longs;o C appiccato in D, & <lb/>&longs;iano due po&longs;&longs;anze eguali in BD, che &longs;o&longs;tengano il pe&longs;o </s> <s>/>Dico, che ogn'vna di que&longs;te po&longs;&longs;anze po&longs;te in BD è vn ter­<lb/>zo del pe&longs;o C. </s> </p> <p type="main"> <s><emph type="italics"/>Hor percioche vna delle due po&longs;&longs;anze è collocata in D, & il pe&longs;o C &longs;tà appiccato <lb/>all'i&longs;te&longs;&longs;o punto D. </s> <s>La po&longs;&longs;anza in D &longs;o&longs;ienirà la parte del pe&longs;o C, che &longs;arà <lb/>eguale ad e&longs;&longs;a po&longs;&longs;an­<lb/>za D. </s> <s>Per laqual co <lb/>&longs;ala po&longs;&longs;anza in B &longs;o <lb/>&longs;tenirà l'altra parte re <lb/>&longs;tante, laqual parte &longs;a <lb/>rà il doppio <expan abbr="tāto">tanto</expan>, quan <lb/>to è la po&longs;&longs;anza di B, <lb/>e&longs;&longs;endo che il pe&longs;o ver <lb/>&longs;o la po&longs;&longs;anza ha la <lb/>proportione i&longs;te&longs;&longs;a, che<emph.end type="italics"/><lb/> <arrow.to.target n="fig90"></arrow.to.target><lb/><emph type="italics"/>ha AB ad AD: & le po&longs;&longs;anze po&longs;te in BD &longs;ono eguali, adunque la po&longs;­<lb/>&longs;anza, che è in B &longs;o&longs;tenirà il doppio più di quello, che &longs;o&longs;tenirà la po&longs;&longs;anza, che è <lb/>in D. </s> <s>Diuida&longs;i dunque il pe&longs;o C in due parti, l'vna delle quali &longs;ia il doppio del­<lb/>l'altra: ilche &longs;i farà, &longs;e lo diuideremo in tre parti eguali EFG, & all'hora FG <lb/>&longs;arà il doppio di E. </s> <s>Co&longs;i la po&longs;&longs;anzain D &longs;o&longs;tenirà la parte E, & la po&longs;&longs;anza <lb/>in B le altre due parti FG. </s> <s>Ambedue dunque le po&longs;&longs;anze po&longs;te in BD tra <lb/>loro eguali <expan abbr="&longs;o&longs;terrãno">&longs;o&longs;terranno</expan> in&longs;ieme tutto il pe&longs;o C. & perche la po&longs;&longs;anza in D &longs;o&longs;tie­<lb/>ne la parte E, laquale è la terza parte del pe&longs;o C, & ad e&longs;&longs;o è eguale, &longs;arà la po&longs;­<lb/>&longs;anza in D vn terzo del pe&longs;o C: & concio&longs;ia che la po&longs;&longs;anza di B &longs;o&longs;tenga le <lb/>parti FG, la po&longs;&longs;anza dellequali po&longs;ta in B è la metà meno: &longs;arà la po&longs;&longs;anza <lb/>in B all'vna delle parti FG, come alla G eguale. </s> <s>& il G è la terza parte <lb/>del pe&longs;o C. </s> <s>La po&longs;&longs;anza dunque in B &longs;arà il terzo del pe&longs;o C. </s> <s>Cia&longs;cuna delle <lb/>po&longs;&longs;anze dunque in BD è vnterzo del pe&longs;o C, che bi&longs;ognaua dimo&longs;trare.<emph.end type="italics"/></s> </p> <pb pagenum="62"/> <figure id="fig90"></figure> <p type="main"> <s><emph type="italics"/>Et &longs;e fo&longs;&longs;ero due leue AB EF diui&longs;e in due parti eguali in GD, i &longs;o&longs;tegni delle­<lb/>quali fo&longs;&longs;ero AF, & il pe&longs;o C fo&longs;&longs;e appiccato all'vna, & l'altra leua in DG<emph.end type="italics"/><lb/> <arrow.to.target n="fig91"></arrow.to.target><lb/><emph type="italics"/>&longs;i fattamente, però che pe&longs;a&longs;&longs;e egualmente nell'vna, & l'altra: & &longs;o&longs;&longs;ero due po&longs;­<lb/>&longs;anze eguali in BG. </s> <s>Si dimo&longs;trerà con ragione in tutto mede&longs;ima, che ogn'vna <lb/>delle po&longs;&longs;anze po&longs;tein B & G è vn terzo del pe&longs;o C.<emph.end type="italics"/></s> </p> <figure id="fig91"></figure> <p type="head"> <s>PROPOSITIONE V.</s> </p> <p type="main"> <s>Se all'vna & l'altra, di cia&longs;cuna girella di due taglie, l'vna delle <lb/>quali &longs;ia po&longs;ta di &longs;opra, & l'altra di &longs;otto, & legata al pe&longs;o; &longs;a­<lb/>rà condotta intornò la corda, legando vno de'&longs;uoi capi alla <lb/>taglia di &longs;otto, & l'altro &longs;ia tenuto dalla po&longs;&longs;anza, che &longs;o&longs;tie­<lb/>ne il pe&longs;o: &longs;arà la po&longs;&longs;anza vn terzo del pe&longs;o. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A, &longs;ia BCD la girella della taglia legata al pe&longs;o A, il cui centro &longs;ia <lb/>E, & &longs;ia FGH l'altra girella della taglia appiccata di &longs;opra, il cui centro &longs;ia <lb/>K: &longs;ia condotta intorno alle girelle la corda LFGHBCDM, laquale &longs;ia lega <lb/>ta alla taglia di &longs;otto in L; & la po&longs; <lb/>&longs;anza, che &longs;o&longs;tiene il pe&longs;o A &longs;ia in <lb/>M. </s> <s>Dico che la po&longs;&longs;anza in M è vn <lb/>terzo del pe&longs;o A. </s> <s>Siano tirate le li­<lb/>nee FH BD per li centri KE egual <lb/>mente di&longs;tanti dall'orizonte, &longs;i come <lb/>nelle precedenti è detto. </s> <s>Hor percio­<lb/>che la corda FL &longs;o&longs;tiene la taglia di <lb/>&longs;otto, laquale &longs;o&longs;tiene la girella nel &longs;uo <lb/>centro E: &longs;arà la corda di L come <lb/>po&longs;&longs;anza che &longs;o&longs;tiene la girella, tanto <lb/>quanto &longs;e fo&longs;&longs;e in e&longs;&longs;o E centro: & <lb/>la po&longs;&longs;anza di M è come &longs;e &longs;te&longs;&longs;e in<emph.end type="italics"/><lb/> <arrow.to.target n="note221"></arrow.to.target> <emph type="italics"/>D; &longs;i farà dunque DB come leua, il <lb/>cui &longs;o&longs;tegno &longs;arà B: ma il pe&longs;o A, <lb/>come di &longs;opra fù dimo&longs;trato, appicca­<lb/>to in E viene &longs;o&longs;tenuto da due po&longs;­<lb/>&longs;anze, l'vna po&longs;ta in D, & l'altra in <lb/>E. </s> <s>& concio&longs;ia, che nel &longs;o&longs;tenere i <lb/>pe&longs;i &longs;tiano le leue FH BD immobi­<lb/>li, &longs;e li pe&longs;i &longs;aranno appiccati alle cor-<emph.end type="italics"/><lb/> <arrow.to.target n="note222"></arrow.to.target> <emph type="italics"/>de FL HB &longs;aranno que&longs;ti i&longs;te&longs;&longs;i egua <lb/>li, per hauere laleua FH il &longs;o&longs;tegno <lb/>nel mezo; altramente dall'vna delle <lb/>parti &longs;i farebbe il mouimento à ba&longs;&longs;o, <lb/>co&longs;a che tuttauia non accade; Adun­<lb/>que tanto &longs;o&longs;tiene la corda FL, quan <lb/>to la HB. </s> <s>Di più percioche dal me­<lb/>zo della leua BD il pe&longs;o pende at­<lb/>taccato, però &longs;e fo&longs;&longs;ero due po&longs;&longs;anze <lb/>in BD che &longs;o&longs;tene&longs;&longs;ero il pe&longs;o, &longs;areb­<lb/>bon fra loro eguali: & benche la cor-<emph.end type="italics"/><lb/> <arrow.to.target n="note223"></arrow.to.target> <emph type="italics"/>da FL &longs;o&longs;tenga e&longs;&longs;a ancora il pe&longs;o, <lb/>poiche ella &longs;ta in loc<gap/> de la po&longs;&longs;anza<emph.end type="italics"/><lb/> <arrow.to.target n="note224"></arrow.to.target> <emph type="italics"/>E, nondimeno percioche &longs;o&longs;tiene da <lb/>quel mede&longs;imo punto, doue è appicca­<lb/>to il pe&longs;o, non farà però che le po&longs;-<emph.end type="italics"/><lb/> <arrow.to.target n="fig92"></arrow.to.target><lb/><emph type="italics"/>&longs;anze, lequali &longs;ono in BD non &longs;iano traloro eguali, peroche aiuta tanto all'v­<lb/>na, quanto all'altra. </s> <s>Ma le po&longs;&longs;anze che &longs;ono in BD &longs;ono le i&longs;te&longs;&longs;e, come &longs;<gap/><emph.end type="italics"/> <pb pagenum="63"/><emph type="italics"/>fu&longs;&longs;ero in HM. </s> <s>Per laqual co&longs;a tanto &longs;o&longs;terrà la corda MD quanto la HB: ma <lb/>co&longs;i &longs;o&longs;tiene HB come FL; adunque la corda MD co&longs;i &longs;o&longs;tenirà, come FL, <lb/>cioè come &longs;e in D & in L fo&longs;&longs;ero appiccati pe&longs;i eguali. </s> <s>Concio&longs;ia co&longs;a dunque, <lb/>che pe&longs;i eguali &longs;ian &longs;o&longs;tenuti da po&longs;&longs;anze vguali, le po&longs;&longs;anze in ML &longs;aranno egua <lb/>li, delle quali è in tutto vna ragione iste&longs;&longs;a, come &longs;e ambedue fo&longs;&longs;ero in DE. <lb/>Onde, e&longs;&longs;endo che il pe&longs;o A &longs;tia attaccato nel mezo della leua BD, & che due <lb/>po&longs;&longs;anze po&longs;te in DE &longs;o&longs;tenente il pe&longs;o &longs;iano eguali: &longs;arà B il &longs;o&longs;tegno, & <lb/>cia&longs;cheduna po&longs;&longs;anza po&longs;ta in DE ouero in ML &longs;arà vn terzo del pe&longs;o A. </s> <s><lb/>Adunquela po&longs;&longs;anza in M &longs;o&longs;tenente il pe&longs;o &longs;arà vn terzo del pe&longs;o A. che<emph.end type="italics"/> <arrow.to.target n="note225"></arrow.to.target><lb/><emph type="italics"/>bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note221"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note222"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note223"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>co rollario di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note224"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo della leua.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note225"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig92"></figure> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Da que&longs;to è manife&longs;to, che ogn'vna delle corde MD FL HB <lb/>&longs;o&longs;tiene la terza parte del pe&longs;o A. </s> </p> <p type="main"> <s><emph type="italics"/>Oltre à ciò &longs;e da M &longs;ar à la corda portata intor­<lb/>no ad vn'altra girella po&longs;ta più &longs;u nella ta­<lb/>glia, che &longs;imilmente &longs;ia attaccata di &longs;opra, il <lb/>cui centro &longs;ia N &longs;i fattamente che peruen <lb/>gain O, & iui &longs;ia tenuta dalla po&longs;&longs;anza; &longs;a <lb/>rà la po&longs;&longs;anza che in O &longs;o&longs;tiene il pe&longs;o A <lb/>parimente vn terzo del pe&longs;o. </s> <s>Percioche la <lb/>corda MD &longs;o&longs;tiene tanto di pe&longs;o, come &longs;e in <lb/>D fo&longs;&longs;e appiccato il pe&longs;o eguale alla terza<emph.end type="italics"/><lb/> <arrow.to.target n="note226"></arrow.to.target> <emph type="italics"/>parte del pe&longs;o A, alla quale è pari la po&longs;­<lb/>&longs;anza in O ad e&longs;&longs;a eguale, cioè vnterzo del <lb/>pe&longs;o A. </s> <s>La po&longs;&longs;anza dunque in O è vn <lb/>terzo del pe&longs;o A.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note226"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et accioche non &longs;i ritorni à dire &longs;pe&longs;&longs;e volte il <lb/>mede&longs;imo, egli fà me&longs;tiero &longs;apere, che lapo&longs; <lb/>&longs;anza in O è &longs;empre eguale à quella, che <lb/>&longs;tain M. come &longs;arebbe à dire, &longs;e la po&longs;&longs;an­<lb/>zain M fo&longs;&longs;e vn quarto, ouero vn quinto, <lb/>ò &longs;imile co&longs;a di e&longs;&longs;o pe&longs;o, la po&longs;&longs;anza parimen <lb/>te in O &longs;arà vn quarto, ouero vn quinto, <lb/>& co&longs;i di mano in mano dell'i&longs;te&longs;&longs;o pe&longs;o, nel <lb/>modo che è di&longs;po&longs;ta la po&longs;&longs;anza di M.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s>Potrebbbe for&longs;e alcuno dubitare in alcune dimo&longs;trationi delle taglie come in que&longs;ta <lb/>quinta p<gap/>opo&longs;itione, tolta da me per e&longs;&longs;empio per e&longs;&longs;ere piu &longs;chietta delle altre, <lb/>che in fatto con la e&longs;perientia non riu&longs;ci&longs;&longs;ero in proportione le forze a' pe&longs;i, co- <pb pagenum="64"/>mela ragione dimo&longs;tra; peroche pre&longs;up ponendo &longs;i nelle dimo&longs;trationi matemati <lb/>che le linee &longs;enza larghezza, & profondità, & co&longs;i le altre co&longs;e imaginando &longs;i &longs;e­<lb/>parate dalla materia, ageuolmente &longs;i per&longs;uadiamo e&longs;&longs;ere vere come dicono. </s> <s>Ma <lb/>la e&longs;perientia poi molte volte mo&longs;tra diuer&longs;ità, & &longs;i trouiamo ingannati, facendo <lb/>la materia gran demente variare le co&longs;e. </s> <s>In que&longs;ta propo&longs;itione &longs;i narra, che rauol <lb/>gendo d'intorno à due girelle di due taglie vna corda, & quel che &longs;egue, la forza <lb/>&longs;arà vn terzo del pe&longs;o, cioè &longs;e il pe&longs;o &longs;ara trecento, egli verrà &longs;o&longs;tenuto dalla po&longs; <lb/>&longs;anza di cento. </s> <s>Direbbe alcuno ciò e&longs;&longs;ere dubbio&longs;o, peroche le girelle, gli a&longs;&longs;etti <lb/>&longs;uoi, le funi, & il pe&longs;o della taglia di &longs;otto fanno re&longs;i&longs;tenza alla forza, & grauano <lb/>sì, che ella non potrà &longs;o&longs;tenere il pe&longs;o. </s> <s>Si ri&longs;ponde che que&longs;te co&longs;e ben farebbo­<lb/>no re&longs;i&longs;tenza nel mouere il pe&longs;o, ma non già nel &longs;o&longs;tentarlo: & bi&longs;ogna notaro <lb/>con diligenza che l'autore in que&longs;te dimo&longs;trationi parla &longs;empre del &longs;o&longs;tenere &longs;o­<lb/>lamente con le forze i pe&longs;i che non calino al ba&longs;&longs;o, non del mouere. </s> <s>Però con­<lb/>&longs;ideri&longs;i, che quando li pe&longs;i &longs;i hanno da far mouere con le po&longs;&longs;anze, allhora le gi­<lb/>relle, & gli altri impedimenti faranno re&longs;i&longs;tenza; ma quando &longs;i ha da far &longs;olamen­<lb/>te che il pe&longs;o &longs;tia fermo, & habbia il &longs;uo contrape&longs;o &longs;emplicemente &longs;enza porre <lb/>in con&longs;ideratione altri ri&longs;petti, che è officio della po&longs;&longs;anza &longs;o&longs;tenente; all'horz <lb/>nè le girelle, nè altro danno re&longs;i&longs;tenza veruna, & la proua fondata &longs;u la ragione <lb/>torna &longs;empre per eccellentia, anzi pare che quanto piu re&longs;i&longs;tenza vi &longs;ia, tanto piu <lb/>facilmente la forza &longs;o&longs;tenga. </s> <s>Auertendo con tutto ciò, che nel fare la e&longs;perienza <lb/>bi&longs;ogna hauere riguardo alla taglia di &longs;otto, & alla corda, lequali hanno la &longs;ua <lb/>grauezza &longs;i fattamente, che &longs;e il pe&longs;o come nell'e&longs;&longs;empio propo&longs;to, &longs;arà trecento <lb/>libre, & la forza cento, & la taglia di &longs;otto con la &longs;ua fune quattordici, è me&longs;tieri <lb/>che alla po&longs;&longs;anza di M &longs;i aggiungano quattro libre, & due terzi di forza, ac­<lb/>cioche po&longs;&longs;a &longs;o&longs;tenere tutto il pe&longs;o, & co&longs;i verrà ad e&longs;&longs;ere in M po&longs;&longs;anza vn ter­<lb/>zo giu&longs;tamente del pe&longs;o. </s> <s>Ma per &longs;apere quanta forza bi&longs;ogni aggiungere alla po&longs; <lb/>&longs;anza, accioche per ri&longs;petto alla taglia di &longs;otto, & alla fune, &longs;o&longs;tenghi il pe&longs;o tut­<lb/>to, faccia&longs;i que&longs;ta ragione. </s> <s>La taglia di &longs;otto con parte della fune, per gratia di <lb/>e&longs;&longs;empio, è quattordici libre, il pe&longs;o è trecento, & la po&longs;&longs;anza cento. </s> <s>Hor per <lb/>la regola detta del tre. </s> <s>Se trecento danno cento, che daranno quattordici? </s> <s>Tro­<lb/>ueran&longs;i quattro libre, & due terzi da e&longs;&longs;ere aggiunte alla po&longs;&longs;anza di M, per <lb/>&longs;o&longs;tenere il pe&longs;o A. </s> <s>“Laqual co&longs;a tocca in &longs;o&longs;tanza l'auttore più à ba&longs;&longs;o <lb/>dicendo.&longs;rdquo; & &longs;i come habbiamo ciò con&longs;iderato nella decimaquinta, & quel, <lb/>che &longs;egue. </s> <s>ilqual loco bi&longs;ogna intendere in que&longs;ta maniera, che le taglie non <lb/>&longs;i deuono pigliare ad vn'i&longs;te&longs;&longs;o modo &longs;empre, ma diuer&longs;amente, come graua­<lb/>no, ilche na&longs;ce dall'e&longs;&longs;ere in vari luoghi, & le po&longs;&longs;anze, & i pe&longs;i collocati, & fer­<lb/>mate le taglie. </s> <s>Hor nella &longs;econda propo&longs;itione di que&longs;to trattato has&longs;i da inten­<lb/>dere la po&longs;&longs;anza e&longs;&longs;ere la meta meno del pe&longs;o, prendendo perlo pe&longs;o, & il pe&longs;o, <lb/>& la taglia di &longs;otto in&longs;ieme, à cui &longs;tà attaccato, come &longs;i vede chiaro nella dimo&longs;tra <lb/>tione della detta &longs;econda propo&longs;itione, doue &longs;i proua che la po&longs;&longs;anza &longs;o&longs;tiene la gi <lb/>rella, laquale &longs;o&longs;tiene anche il re&longs;to della taglia nell'a&longs;&longs;etto, alla qual taglia è attac­<lb/>cato il pe&longs;o, oue &longs;i cono&longs;ce e&longs;pre&longs;&longs;o, che la taglia, & il pe&longs;o s'hanno à pigliare <lb/>per tutto il pe&longs;o. </s> <s>Per la qual co&longs;a, &longs;e in quel ca&longs;o il pe&longs;o in&longs;ieme con la taglia pe­<lb/>&longs;eranno vinti, la po&longs;&longs;anza che gli &longs;o&longs;tenterà &longs;arà dieci. </s> <s>Et per vn'altro e&longs;&longs;empio <lb/>nella nona propo&longs;<gap/>ione di que&longs;to nel primo ca&longs;o, &longs;e il pe&longs;o con la taglia di &longs;otto <lb/>pe&longs;eranno vinticinque, la po&longs;&longs;anza &longs;o&longs;tenente &longs;arà cinque. </s> <s>& co&longs;i egli è me&longs;t<gap/>eri <lb/>hauer con&longs;ideratione nelle altre, cioè di&longs;tinguere doue è la grauezza della taglia, <lb/>quando graua di &longs;otto &longs;olamente, come nelle allegate propo&longs;itioni, & &longs;imili: & <lb/>quando &longs;olamente di &longs;opra, come nelle propo&longs;itioni 17. & 18. & &longs;imili: & quan <lb/>do ambedue le taglie grauano di &longs;opra, & di &longs;otto, come nelle propo&longs;itioni 20. <lb/>22. & 23. & &longs;imili: & quando anche nel'vna taglia, ne l'altra grauano, come nella <lb/>prima propo&longs;itione & nella 19. anzi in e&longs;&longs;a 19. la taglia di &longs;otto aiuta la <expan abbr="po&longs;sãza">po&longs;sanza</expan> ad <lb/>e&longs;&longs;ere piu leggiera: & nel &longs;econdo ca&longs;o dopo il corollario della 16. propo&longs;itione, <lb/>& &longs;imili. </s> <s>& oltre à ciò deue&longs;i por mente alle corde ancora, la grauezza delle qua­<lb/>li non hà &longs;empre da e&longs;&longs;ere con&longs;iderata, peroche grauano nelle propo&longs;itioni 15. 17. <lb/>ma non grauano già nella 19. </s> </p> <p type="main"> <s>Ne parmi etiandio che &longs;i habbia ad hauere punto di riguardo alla picciolezza, & <lb/>grandezza delle girelle po&longs;te nelle taglie, & de gli a&longs;&longs;etti &longs;uoi, credendo che per <lb/>neces&longs;ità habbiano da e&longs;&longs;ere lauorati con mi&longs;ura tale, & proportione co&longs;i ac<gap/>u­<lb/>rata, che mancando da quella non rie&longs;cano le dimo&longs;trationi alla e&longs;perientia; per <lb/>roche, &longs;i come nota l'autore poco appre&longs;&longs;o, ba&longs;ta che con certa conueneuole mi&longs;n <lb/>ra, & proportione le girelle nelle taglie &longs;iano maggiori l'vna dell'altra &longs;i fattamen <lb/>te, che le corde non &longs;i to cchino, & freghino fra loro, & co&longs;i vengano ad impedi <lb/>re i mouimenti delle po&longs;&longs;anze, & de' pe&longs;i. </s> </p> <p type="head"> <s>PROPOSITIONE VI.</s> </p> <p type="main"> <s>Siano due leue AB CD diui&longs;e in due parti eguali in EF, li <lb/>&longs;o&longs;tegni delle quali &longs;iano in BD; & &longs;ia il pe&longs;o G in EF ap <lb/>piccato all'vna, & l'altra leua &longs;i fattamente, che pe&longs;i dall'vna, <lb/>& dall'altra egualmente: & &longs;iano due po&longs;&longs;anze in AC egua­<lb/>li, che &longs;o&longs;tengano il pe&longs;o. </s> <s>Dico, che ogn'vna delle po&longs;&longs;anze <lb/>in AC è vn quarto del pe&longs;o G. </s> </p> <p type="main"> <s> <arrow.to.target n="note227"></arrow.to.target> <emph type="italics"/>Concio&longs;ia che le po&longs;&longs;anze po­<lb/>&longs;te in AC &longs;o&longs;tengano tut <lb/>to il pe&longs;o G, & la po&longs;&longs;an­<lb/>za di A ver&longs;ola parte del <lb/>pe&longs;o, che &longs;o&longs;tiene, &longs;ia come <lb/>BE à BA, & la po&longs;&longs;an­<lb/>zain C alla parte di e&longs;&longs;o <lb/>G pe&longs;o &longs;o&longs;tenuto da lei &longs;ia <lb/>co&longs;i, come DF à DC, & <lb/>come BE à BA, co&longs;i è <lb/>DF à DC: &longs;ar à lapo&longs;&longs;an <lb/>za po&longs;ta in A ver&longs;o la par <lb/>te del pe&longs;o, che &longs;o&longs;tiene, co-<emph.end type="italics"/><lb/> <arrow.to.target n="fig93"></arrow.to.target><lb/><emph type="italics"/>me la po&longs;&longs;anza di C ver&longs;o la parte di e&longs;&longs;o pe&longs;o, che &longs;o&longs;tiene: & le po&longs;&longs;anze po&longs;ie <lb/>in AC &longs;ono eguali; &longs;aranno dunque le parti del pe&longs;o G eguali, lequali &longs;ono &longs;o-<emph.end type="italics"/> <pb pagenum="65"/><emph type="italics"/>&longs;tenute dalle po&longs;&longs;anze. </s> <s>Per laqual co&longs;a cia&longs;cuna po&longs;&longs;anza po&longs;ta in AC &longs;o&longs;terrà <lb/>la metà del pe&longs;o G. </s> <s>Mala po&longs;&longs;anza in A è la metà meno del pe&longs;o, che &longs;o&longs;tie­<lb/>ne; adunque la po&longs;&longs;anza in A &longs;arà per lo mezo della metà, cioè eguale alla quar <lb/>ta portione del pe&longs;o G; & però &longs;arà il quarto del pe&longs;o G, nè altramente &longs;i di­<lb/>mo&longs;trerà la po&longs;&longs;anza in C e&longs;&longs;ere vn quarto dell'i&longs;te&longs;&longs;o pe&longs;o G. che bi&longs;ognaua <lb/>mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note227"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo nella leua.<emph.end type="italics"/></s> </p> <figure id="fig93"></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e &longs;aranno tre leue AB <lb/>CD EF diui&longs;e in due <lb/>parti eguali in GHK, li <lb/>&longs;o&longs;tegni delle quali &longs;iano <lb/>BDF, & il pe&longs;o L &longs;ia <lb/>nell'i&longs;te&longs;&longs;o modo appicca­<lb/>to in GHK: & &longs;iano <lb/>tre po&longs;&longs;anze in ACE <lb/>eguali, che &longs;o&longs;tengano il <lb/>pe&longs;o: &longs;i mo&longs;trerà &longs;imil­<lb/>mente cia&longs;cuna po&longs;&longs;anza <lb/>e&longs;&longs;ere vn &longs;e&longs;to del pe&longs;o <lb/>L: & con questo ordi­<lb/>ne &longs;e fo&longs;&longs;ero quattro le­<lb/>ue, & quattro po&longs;&longs;anze, <lb/>cia&longs;cuna po&longs;&longs;anza &longs;arà <lb/>la ottaua parte del pe&longs;o, & co&longs;i di mano in mano in infinito.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>PROPOSITIONE VII.</s> </p> <p type="main"> <s>Se à tre girelle di due taglie, l'vna delle quali po&longs;ta di &longs;opra hab <lb/>bia vna &longs;ola girella, & l'altra di &longs;otto ne habbia due, & &longs;ia lega <lb/>ta al pe&longs;o; &longs;ia po&longs;ta d'intorno la corda; legando l'vn de' capi <lb/>&longs;uoi in qualche loco, & l'altro &longs;ia tenuto dalla po&longs;&longs;anza, che <lb/>&longs;o&longs;tiene il pe&longs;o. </s> <s>La po&longs;&longs;anza &longs;arà vn quarto del pe&longs;o. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A: &longs;iano le tre girelle, il centro dellequali &longs;ia BCD: & la girella, il <lb/>cui centro è D, &longs;ia della taglia appiccata di <lb/>&longs;opra: ma quelle girelle, il cui centro è in B <lb/>C &longs;iano della taglia legata al pe&longs;o A: & <lb/>la corda EFGHKLNOP &longs;ia condotta <lb/>intorno à tutte le girelle, & legata in E: & <lb/>&longs;ia la forza che &longs;o&longs;tiene il pe&longs;o A in P. </s> <s><lb/>Dicola po&longs;&longs;anza in P e&longs;&longs;ere vn quarto del <lb/>pe&longs;o A. </s> <s>Siano tirate le linee KL GF ON <lb/>per li centri delle girelle, &longs;i che &longs;iano egual­<lb/>mente di&longs;tanti dall'orizonte; le quali per le co <lb/>&longs;e, che già &longs;ono dette, &longs;aranno come leue. </s> <s>& <lb/>percioche per cagione della leua, ouero bilan­<lb/>cia KL, il cui &longs;o&longs;tegno, ouero centro è nel<emph.end type="italics"/><lb/> <arrow.to.target n="note228"></arrow.to.target> <emph type="italics"/>mezo, tanto &longs;o&longs;tiene la corda KG, quanto <lb/>la NL non &longs;i &longs;acendo mouimento in niu­<lb/>na delle parti: Di più per cau&longs;a della leua <lb/>GF dal cui mezo, come &longs;o&longs;pe&longs;o dipende il <lb/>pe&longs;o; &longs;e fo&longs;&longs;ero due po&longs;&longs;anze in GF, oue-<emph.end type="italics"/><lb/> <arrow.to.target n="note229"></arrow.to.target> <emph type="italics"/>ro in HE, (percioche &longs;i come è &longs;tato più <lb/>volte detto, la ragione dell'vno, & dell'al­<lb/>tro &longs;ito è pari) &longs;arebbono per certo que&longs;te <lb/>tali po&longs;&longs;anze eguali fraloro. </s> <s>Onde co&longs;i &longs;o­<lb/>&longs;tiene la corda HG, come EF: &longs;imilmen <lb/>te &longs;imo&longs;trerà tanto &longs;o&longs;tenere la corda PO, <lb/>quanto la NL. </s> <s>Per laqual co&longs;a le corde <lb/>PO KG EF LN &longs;o&longs;tengono egualmen­<lb/>te. </s> <s>Adunque &longs;o&longs;tiene egualmente sì la cor­<lb/>da PO, come la KG. </s> <s>Se dunque s'inten­<lb/>de&longs;&longs;ero e&longs;&longs;ere due po&longs;&longs;anze in OG, ouero in <lb/>PH, che è il mede&longs;imo, lequali tuttauia &longs;o­<lb/>&longs;tenghino il pe&longs;o, come &longs;o&longs;tengono le corde, <lb/>&longs;arebbono per certo eguali: & GF ON <lb/>baurebbono le &longs;orze di due leue, il &longs;o&longs;tegno <lb/>delle quali &longs;aranno FN & il pe&longs;o A &longs;a <lb/>rà appiccato in BC, che è il mezo delle le­<lb/>ue. </s> <s>& percioche tutte le corde &longs;o&longs;tengo­<lb/>no egualmente, tanto &longs;o&longs;teniranno le due<emph.end type="italics"/><lb/> <arrow.to.target n="fig94"></arrow.to.target><lb/><emph type="italics"/>PO LN quanto le due KG EF. </s> <s>tanto dunque &longs;o&longs;terrà la leua ON, quan-<emph.end type="italics"/><lb/> <arrow.to.target n="note230"></arrow.to.target> <emph type="italics"/>to la leua GF. </s> <s>Onde nell'vna, & l'altra leua ON GF pe&longs;erà egualmente il <lb/>pe&longs;o. </s> <s>&longs;arà dunque ogni po&longs;&longs;anza che è in PH vn quarto del pe&longs;o A. & e&longs;&longs;en<emph.end type="italics"/> <pb pagenum="66"/><emph type="italics"/>do, che la corda KG &longs;i prenda in loco di po&longs;&longs;anza, come quella, che non &longs;o&longs;tiene <lb/>altramente di quel che faccia PO, &longs;arà la po&longs;&longs;anza di P, che &longs;o&longs;tiene il pe&longs;o A <lb/>vn quarto di e&longs;&longs;o pe&longs;o. </s> <s>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note228"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note229"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 2. <emph type="italics"/>co rollario della<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note230"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig94"></figure> <p type="head"> <s>COROLLARIO I.</s> </p> <p type="main"> <s>Di qui è manife&longs;to, che cia&longs;cuna corda EF GK LN OP &longs;o­<lb/>&longs;tiene la quarta parte del pe&longs;o A. </s> </p> <p type="head"> <s>COROLLARIO II.</s> </p> <p type="main"> <s>E chiaro ancora, che non meno &longs;o&longs;tiene la girella il cui centro <lb/>è C, di quello che faccia la girella, il centro dellaquale è B. </s> </p> <pb/> <p type="head"> <s>Altramente.</s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;te ancora le co&longs;e mede&longs;ime, &longs;e fo&longs;&longs;ero <lb/>due po&longs;&longs;anze eguali, che &longs;o&longs;tene&longs;&longs;ero<emph.end type="italics"/><lb/> <arrow.to.target n="note231"></arrow.to.target> <emph type="italics"/>il pe&longs;o A, l'vna in O, & l'altrain <lb/>C: &longs;arebbe cia&longs;cuna delle dette po&longs;&longs;in­<lb/>ze vnterzo del pe&longs;o A. </s> <s>Ma perche <lb/>la leua GF, il cui &longs;o&longs;tegno è F, è <lb/>diui&longs;a in due parti eguali nel C. &longs;e dun <lb/>que &longs;i porrà la po&longs;&longs;anza in G che &longs;o­<lb/>&longs;tenga l'i&longs;te&longs;&longs;o pe&longs;o, come la po&longs;&longs;anza <lb/>di C, &longs;arà la po&longs;&longs;anza di G la metà <lb/>della po&longs;&longs;anza, che fo&longs;&longs;e in C; per­<lb/>cioche &longs;e la po&longs;&longs;anza di C per &longs;e &longs;te&longs;&longs;a <lb/>&longs;o&longs;tene&longs;&longs;e il pe&longs;o, che è appiccato in C, <lb/>&longs;arebbe per certo eguale ad e&longs;&longs;o pe&longs;o; et <lb/>&longs;e l'i&longs;te&longs;&longs;o pe&longs;o fo&longs;&longs;e &longs;o&longs;tenuto dalla po&longs; <lb/>&longs;anza di G, &longs;arebbe il doppio di e&longs;&longs;a G <lb/>po&longs;&longs;anza, & la po&longs;&longs;anza di C &longs;areb-<emph.end type="italics"/><lb/> <arrow.to.target n="note232"></arrow.to.target> <emph type="italics"/>be vn terzo del pe&longs;o A; dunque la <lb/>po&longs;&longs;anza di G &longs;arebbe vn &longs;e&longs;to della <lb/>po&longs;&longs;anza del pe&longs;o A. </s> <s>Per laqual co <lb/>&longs;a, e&longs;&longs;endo, che la po&longs;&longs;anza di O &longs;ia vn <lb/>terzo del pe&longs;o A, & la po&longs;&longs;anza di <lb/>G vn &longs;e&longs;to: &longs;ara l'vna, & l'altra po&longs;­<lb/>&longs;anza in&longs;ieme po&longs;te in OG la metà <lb/>del pe&longs;o A, percioche la terza par­<lb/>te con la &longs;e&longs;ta &longs;à la metà. </s> <s>Ma per­<lb/>cioche la po&longs;&longs;anza di OG, ouero di <lb/>PH, (come prima è detto) &longs;ono fra <lb/>loro eguali, & l'vna, & l'altra in&longs;ie­<lb/>me &longs;ono la metà del pe&longs;o A, &longs;arà <lb/>ogn'vna delle po&longs;&longs;anze po&longs;te in PH <lb/>vn quarto di e&longs;&longs;o A. </s> <s>Adunque la <lb/>po&longs;&longs;anza di P che &longs;o&longs;tiene il pe&longs;o A <lb/>&longs;arà vn quarto di e&longs;&longs;o pe&longs;o A. che era <lb/>da mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note231"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note232"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo della leua.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="67"/> <p type="main"> <s><emph type="italics"/>Ma &longs;e'la corda <lb/>&longs;arà legata in <lb/>E, & &longs;ia <lb/>dauantaggio <lb/>inuolta intor <lb/>no à quattro <lb/>girelle, et per <lb/>uenga in P, <lb/>&longs;imo&longs;trerà &longs;i <lb/><expan abbr="milmēte">milmente</expan>, che <lb/>la po&longs;&longs;anza <lb/>di P &longs;arà <lb/>vn quarto <lb/>del pe&longs;o A; <lb/>peroche egli <lb/>è il mede&longs;i­<lb/>mo, come &longs;e <lb/>la corda fo&longs;­<lb/>&longs;e legata in <lb/>L, & che la <lb/>po&longs;&longs;anza &longs;o­<lb/>&longs;tene&longs;&longs;e il pe <lb/>&longs;o con la cor­<lb/>da inuolta in <lb/>torno à tre gi <lb/>relle &longs;olamen <lb/>te, i centri <lb/>delle quali fo&longs; <lb/>&longs;ero BCQ, <lb/>percioche la <lb/>girella, il cui <lb/>centro è D, <lb/>del tutto è <lb/>inutile.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>PROPOSITIONE VIII.</s> </p> <p type="main"> <s>Siano due leue AB CD diui&longs;e in due parti eguali EF, i &longs;o­<lb/>&longs;tegni delle quali &longs;iano AC, & &longs;ia appiccato il pe&longs;o G ne' <lb/>punti EF all'vna, & l'altra leua, &longs;i fattamente, che dall'vno, <lb/>& l'altro pe&longs;i egualmente: & &longs;iano tre po&longs;&longs;anze eguali in BD <lb/>E che &longs;o&longs;tenghino il pe&longs;o G. </s> <s>Dico, che cia&longs;cuna delle det­<lb/>te po&longs;&longs;anze &longs;eparatamente è vn quinto del pe&longs;o G. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche il pe&longs;o G &longs;ta appiccato in EF, & &longs;ono le tre po&longs;&longs;anze in EBD egua­<lb/>li: però la po&longs;&longs;anza di E &longs;o&longs;terrà la parte &longs;olamente del pe&longs;o G, che &longs;arà eguale <lb/>ad e&longs;&longs;a po&longs;&longs;anza di E, ma <lb/>le po&longs;&longs;anze di BD &longs;o&longs;terran <lb/>no la parte re&longs;tante, & la<emph.end type="italics"/><lb/> <arrow.to.target n="note233"></arrow.to.target> <emph type="italics"/>parte, che è da B &longs;o&longs;tenu­<lb/>ta, &longs;arà il doppio di e&longs;&longs;o: ma <lb/>la parte &longs;o&longs;tenuta da D &longs;a­<lb/>rà &longs;imilmente il doppio di e&longs; <lb/>&longs;o D per cau&longs;a della pro­<lb/>portione di BA ver&longs;o AE, <lb/>& di DC ver&longs;o CF. </s> <s>Con <lb/>cio&longs;ia dunque, che le po&longs;&longs;an-<emph.end type="italics"/><lb/> <arrow.to.target n="note234"></arrow.to.target> <emph type="italics"/>ze di BD &longs;iano eguali, &longs;a­<lb/>ranno anche (per quel che di <lb/>&longs;opra è detto) le parti del pe<emph.end type="italics"/><lb/> <arrow.to.target n="fig95"></arrow.to.target><lb/><emph type="italics"/>&longs;o G, lequali &longs;ono &longs;o&longs;tenute dalle po&longs;&longs;anze di BD, &longs;ra loro eguali, & ogni vna <lb/>&longs;ar à il doppio di quella tal parte, che è &longs;o&longs;tenuta dalla po&longs;&longs;anza di E. </s> <s>Diuida&longs;i <lb/>dunque il pe&longs;o G in tre parti, delle quali due &longs;iano fra loro eguali, & di più ogni <lb/>vna di loro &longs;eparatamente &longs;ia il doppio dell'altra terza parte, ilche accaderà, &longs;e <lb/>in cinque parti eguali HKLMN &longs;arà diui&longs;o: percioche la parte compo&longs;ta di due <lb/>parti KL è il doppio della parte H, & la parte ancora di MN è &longs;imilmen­<lb/>te il doppio della parte i&longs;te&longs;&longs;a H. </s> <s>Per laqual co&longs;a anche la parte KL &longs;arà egua­<lb/>le alla parte MN. </s> <s>Ma &longs;o&longs;tenga la po&longs;&longs;anza di E la parte di H; & la po&longs;&longs;an <lb/>za di B le parti di KL: & la po&longs;&longs;anza di D le parti MN; adunque le tre <lb/>po&longs;&longs;anze eguali po&longs;te in BDE &longs;o&longs;terranno tutto il pe&longs;o G: & ogn'vna delle <lb/>po&longs;&longs;anze di BD &longs;o&longs;terrà il doppio di quel che &longs;o&longs;tiene la po&longs;&longs;anza di E. </s> <s>Però <lb/>e&longs;&longs;endo che la po&longs;&longs;anza di E &longs;o&longs;tenga la parte di H, laquale è la quinta parte del <lb/>pe&longs;o G, & &longs;ia ad e&longs;&longs;o eguale, &longs;arà la po&longs;&longs;anza di E vn quinto del pe&longs;o G. & <lb/>percioche la po&longs;&longs;anza di B &longs;o&longs;tiene le parti di KL, lequali &longs;ono il doppio & del-<emph.end type="italics"/> <pb pagenum="68"/><emph type="italics"/>la po&longs;&longs;anza di B, & della parte di H, &longs;arà ancora la po&longs;&longs;anza di B ad e&longs;&longs;o H <lb/>eguale. </s> <s>Per laqual co&longs;a &longs;arà vn quinto del pe&longs;o G. </s> <s>Ne altrimente &longs;i dimo&longs;tre­<lb/>rà, che la po&longs;&longs;anza di D è vn quinto del pe&longs;o G. cia&longs;cuna po&longs;&longs;anza dunque in <lb/>BDE è vn quinto del pe&longs;o G. che bi&longs;ognaua dimo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note233"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>di questa nella leua.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note234"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig95"></figure> <figure></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e &longs;aranno tre leue AB <lb/>CD EF diui&longs;e in due <lb/>parti eguali in GHK, i <lb/>&longs;o&longs;tegni dellequali &longs;iano A <lb/>CE, & il pe&longs;o L nel mo <lb/>do i&longs;te&longs;&longs;o &longs;ia appiccato in <lb/>GHK, & &longs;iano quattro <lb/>po&longs;&longs;anze eguali in BD <lb/>FG che &longs;o&longs;tengano il pe­<lb/>&longs;o L; &longs;i mo&longs;trerà con &longs;imi­<lb/>gliante modo, che cia&longs;cuna <lb/>po&longs;&longs;anza in BD FG &longs;a­<lb/>rà vn &longs;ettimo del pe&longs;o L: <lb/>& &longs;e quattro fo&longs;&longs;ero le le­<lb/>ue, & cinque le po&longs;&longs;anze <lb/>eguali &longs;o&longs;tenenti il pe&longs;o; con l'i&longs;te&longs;&longs;o modo ancora &longs;i mo&longs;trerebbe che ogni vna del­<lb/>le po&longs;&longs;anze &longs;arebbe vn nono del pe&longs;o, & co&longs;i di mano in mano &longs;ucce&longs;&longs;iuamente.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE IX.</s> </p> <p type="main"> <s>Se à quattro girelle di due taglie, l'vna delle quali &longs;ia po&longs;ta di <lb/>&longs;opra, & l'altra di &longs;otto legata al pe&longs;o, &longs;ia condotta intorno <lb/>la corda, legando l'vno de'&longs;uoi capi alla taglia di &longs;otto, & l'al­<lb/>tro &longs;ia ritenuto dalla po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o. </s> <s>&longs;arà la po&longs;­<lb/>&longs;anza vn quinto del pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A, alquale &longs;ia legata <lb/>la taglia, che habbia due girel­<lb/>le, i cui centri &longs;iano BC: & <lb/>&longs;ia la taglia appiccata di &longs;opra, <lb/>che habbia due altre girelle, i <lb/>cui centri &longs;iano DE, & la <lb/>corda &longs;ia tirata intorno à tutte <lb/>le girelle, laquale &longs;ia legata al­<lb/>la taglia di &longs;otto in F: & &longs;ia <lb/>la po&longs;&longs;anza in G che &longs;o&longs;tiene <lb/>il pe&longs;o A. </s> <s>Dico che la po&longs;&longs;an <lb/>za di G è vn quinto del pe&longs;o <lb/>A. </s> <s>Siano tirate le linee HK <lb/>LM per li centri BC egual­<lb/>mente di&longs;tanti dall'orizonte, le <lb/>quali nel modo i&longs;te&longs;&longs;o, che di <lb/>&longs;opra è &longs;tato detto, dimo&longs;trere­<lb/>mo e&longs;&longs;ere come leue, i &longs;o&longs;tegni <lb/>delle quali &longs;ono KM, & il pe <lb/>&longs;o A pende attaccato nel me­<lb/>zo BC dell'vna, & l'altra le­<lb/>ua, & le tre po&longs;&longs;anze LHC, che <lb/>&longs;o&longs;tengono il pe&longs;o, lequali con <lb/>&longs;imile modo mo&longs;treremo e&longs;&longs;ere <lb/>eguali: percioche le corde fanno <lb/>l'i&longs;ie&longs;&longs;o officio, come &longs;e fo&longs;&longs;ero <lb/>po&longs;&longs;anze: & percioche il pe&longs;o <lb/>dall'vna, & l'altra leua HK <lb/>LM pe&longs;a egualmente, ilche &longs;i <lb/>dimo&longs;trerà ancora, come nelle <lb/>precedenti è &longs;tato dimo&longs;trato:<emph.end type="italics"/><lb/> <arrow.to.target n="note235"></arrow.to.target> <emph type="italics"/>&longs;arà ogni po&longs;&longs;anza po&longs;ta sì in L <lb/>ouero in G, che è il mede&longs;imo; <lb/>& sì in H & in C, cioè in F <lb/>vn quinto del pe&longs;o A. </s> <s>La po&longs; <lb/>&longs;anza dunque di G, che &longs;o&longs;tie­<lb/>ne il pe&longs;o A. &longs;arà vn quinto <lb/>di e&longs;&longs;o pe&longs;o A. che bi&longs;ognaua <lb/>mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note235"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="69"/> <p type="main"> <s><emph type="italics"/>Che &longs;e dauantaggio &longs;i traporterà la cor­<lb/>da in F d'intorno ad vn'altra girella, <lb/>il cui centro &longs;ia N, & &longs;ia legata <lb/>in O, &longs;i prouerà &longs;imilmente per due <lb/>ragioni, come nella &longs;ettima propo&longs;i­<lb/>tione di que&longs;to, che la po&longs;&longs;anza di G <lb/>che &longs;o&longs;tiene il pe&longs;o A, è vn &longs;e&longs;to<emph.end type="italics"/> <arrow.to.target n="note236"></arrow.to.target><lb/><emph type="italics"/>di e&longs;&longs;o pe&longs;o A. </s> <s>Percioche prima dal <lb/>le treleue LM HK FP licui &longs;o­<lb/>stegni &longs;ono in KP, & il pe&longs;o è ap­<lb/>piccato nel mezo delle leue, & le tre <lb/>po&longs;&longs;anze po&longs;te in LHF che &longs;o&longs;ten-<emph.end type="italics"/> <arrow.to.target n="note237"></arrow.to.target><lb/><emph type="italics"/>gono il pe&longs;o &longs;ono eguali: poi dalle po&longs; <lb/>&longs;anze di LHN cia&longs;cuna delle quali <lb/>&longs;arebbe vn quinto del pe&longs;o A, per­<lb/>cioche ambedue le po&longs;&longs;anze in&longs;ieme <lb/>po&longs;te in LH &longs;arebbono &longs;otto doppie &longs;e&longs; <lb/>quialtere al pe&longs;o, & la po&longs;&longs;anza di F <lb/>&longs;arebbe vn decimo, e&longs;&longs;endo la metà di <lb/>e&longs;&longs;a N. </s> <s>Ma due quinte parti con <lb/>vna decima parte fanno la metà, la <lb/>qual metà &longs;e &longs;arà diui&longs;a per tre, ri­<lb/>&longs;ponderà la &longs;e&longs;ta parte del pe&longs;o à cia­<lb/>&longs;cuna delle po&longs;&longs;anze po&longs;te in LHF. </s> <s><lb/>Dalle quali co&longs;e è manife&longs;to la po&longs;&longs;an­<lb/>za di G e&longs;&longs;ere vn &longs;e&longs;to del pe&longs;o A; <lb/>& &longs;i dimo&longs;trerà &longs;imilmente che cia­<lb/>&longs;cuna girella &longs;o&longs;tiene eguale portione <lb/>del pe&longs;o.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note236"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>di questo<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note237"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <pb/> <p type="main"> <s>In que&longs;to trattato della taglia, &longs;i come in tutti gli altri an cora, l'autore pre&longs;uppone, <lb/>che qualunque per&longs;ona &longs;i mette à leggere il &longs;uo libro delle Mechaniche &longs;ia inten­<lb/>dente di numeri, & di Geometria, & però ha &longs;empre mantenuto quello accurato <lb/>&longs;tile, & dimo &longs;tratiuo co&longs;tumato da buoni Matematici, v&longs;ando i vocaboli proprij <lb/>della &longs;cienza, alcuni de' quali io hò ben potuto volgarizare facilmente, &longs;i che <lb/>ogn'vno gli po&longs;&longs;a intendere, come per e&longs;&longs;empio, nelle proportioni duplum, tri­<lb/>plum, quadruplum, & gli altri &longs;imili, ponendo in vece loro due volte tanto, tre <lb/>volte tanto, & quattro volte tanto: & co&longs;i per 'oppo&longs;ito &longs;ub duplum, &longs;ubtriplu, <lb/>& &longs;ub quadruplum, la metà, vn terzo, & vn quarto: & parimente &longs;e&longs;quialterum, <lb/>&longs;e&longs;quitertium, & &longs;e&longs;quiquartum, & gli altri &longs;imili, che vogliono dire vna volta & <lb/>meza, vna volta, & vn terzo, & vna volta & vn quarto. </s> <s>Que&longs;ti dico s'hanno po <lb/>tuto ben dire, & facilmente nella no&longs;tra lingua. </s> <s>Ma nell'ampiezza delle propor­<lb/>tioni trouando&longs;i altri vocaboli a&longs;&longs;ai, i quali non è pos&longs;ibile co&longs;i adattare alla no­<lb/>&longs;tra lingua, tra quali alcuni &longs;i trouano po&longs;ti dall'autore in que&longs;to trattato della ta <lb/>glia, & io &longs;ono &longs;tato sforzato à la&longs;ciargli co&longs;i, come erano, per mancamento di pa <lb/>role, che nella no&longs;tra fauella gli po&longs;&longs;ano e&longs;primere; hò giudicato douer e&longs;&longs;ere co <lb/>&longs;a vtile il dichiarare tuttii predetti vocaboli pertinenti alle proportioni, che ha il <lb/>pe&longs;o alla po&longs;&longs;anza, & la po&longs;&longs;anza al pe&longs;o &longs;critti dall'autore in que&longs;to trattato della <lb/>taglia, accio che quelle per&longs;one lequali non po&longs;&longs;edono que&longs;ti termini, non habbia <lb/>no fatica di andare &longs;tudiando iloro &longs;ignificati. </s> </p> <p type="main"> <s>Dico dunque vna quantità poter&longs;i paragonare, & hauere proportione con vn'altra <lb/>in tre modi principali, la&longs;ciando hora le più &longs;ottili di&longs;tintioni. </s> <s>Primieramente <lb/>come maggiore ver&longs;o la minore, dapoi come minore ver&longs;o la maggiore, & in fi­<lb/>ne come eguale ver&longs;o la eguale. </s> <s>Tutta la dottrina delle'proportioni, con&longs;i&longs;te in <lb/>que&longs;ti riguardi, cioè dal maggiore al minore, dal min ore al maggiore, & dall'e­<lb/>quale all'equale. </s> <s>Hor quando vna quantità, che &longs;ia maggiore è paragonata con <lb/>vn'altra, che &longs;ia minore, che &longs;i dice proportione di maggiore di&longs;uguaglianza, na­<lb/>&longs;cono cinque generi di proportioni, l'vno è il moltiphce &longs;chietto, il &longs;econdo è il <lb/>&longs;opraparticolare, il rerzo il &longs;o prapartiente, il quarto il moltiplice &longs;opraparticola­<lb/>re, & il quinto & vltimo il moltiplice &longs;oprapartiente. </s> <s>Ma quando &longs;i fa compara­<lb/>tione della minore quantità ver&longs;o la maggiore, all'hora &longs;i producono cinque altri <lb/>generi oppo&longs;ti apunto à i predetti cinque, & &longs;i dicono di minore di&longs;uguaglian­<lb/>za, à i quali per fargli differenti da loro &longs;i aggiunge da Latini il &longs;ub, cioè &longs;otto, <lb/>&longs;criuendo&longs;t &longs;otto moltiplice, &longs;otto&longs;opra particolare, &longs;otto &longs;oprapartiente, &longs;otto <lb/>moltiplice &longs;opra particolare, & &longs;otto moltiplice &longs;oprapartiente. </s> <s>Tutte le propor­<lb/>tioni dunque &longs;ono compre&longs;e in vniuer&longs;ale da que&longs;ti diece generi oppo&longs;ti fra &longs;e <lb/>l'vn l'altro, cia&longs;cheduno de quali poi ha le &longs;ue &longs;petie differenti di proportioni. </s> <s>Ma <lb/>io non hò qui intentione di numerarle, nè dichiarare diffu&longs;amente que&longs;ta materia <lb/>delle proportioni, ma &longs;olamente li vocaboli po&longs;ti dall'autore nel pre&longs;ente libro <lb/>della taglia, ba&longs;tando mi hauerne dato in generale vna rozza cognitione. </s> <s>Ma chi <lb/>di ciò de&longs;idera hauere intero cono&longs;cimento legga tra i &longs;crittori della lingua Ita­<lb/>liana Fra Luca dal Borgo, il Tartaglia ne i libri della Arithmetica, & il dottis&longs;imo <lb/>Zarlino nella prima parte delle In&longs;titutioni Harmoniche. </s> <s>Dice l'autore in que&longs;to <lb/>loco. </s> <s>Percio che &longs;arebbono ambedue le po&longs;&longs;anze infieme in LH &longs;otto doppie <lb/>&longs;e&longs;quialtere di e&longs;&longs;o pe&longs;o. </s> <s>Cioè le due po&longs;&longs;anze po&longs;te in LH haurebbono quella <lb/>proportione ver&longs;o il pe&longs;o, che ha 2. à 5. cioè &longs;e il pe&longs;o fo&longs;&longs;e come cinque, le po&longs;­<lb/>&longs;anze larebbono come 2. che è la proportione &longs;otto dop pia &longs;e&longs;quialtera. </s> <s>Segue <pb pagenum="70"/>poi, Ma due quinte con vna decima fanno la me <lb/>tà, cioè à &longs;ommare in&longs;ieme due quinti, & vn <lb/>decimo fanno la metà di cinque, pero che li <lb/>due quinti &longs;ono due parti del cinque, & la deci <lb/>ma parte è la metà di vn quinto, tanto che met­<lb/>tono in&longs;ieme due, & mezo, che &longs;ono la metà di <lb/>cin que. </s> <s>Che &longs;e que&longs;ta metà poi &longs;arà diui&longs;a per <lb/><gap/>re, ne riu&longs;cirà la &longs;e&longs;ta parte da e&longs;&longs;ere attribuita à <lb/>cia&longs;cheduna delle tre po&longs;&longs;anze po&longs;te in LHF. </s> <s><lb/>Il modo del diuidere la metà per tre è facile, & <lb/>fas&longs;i in que&longs;ta maniera ponendo tre di &longs;opra, & <lb/>vno di &longs;otto; & vno di &longs;opra, & due di &longs;otto <expan abbr="cõ">com</expan> <lb/>la &longs;ua linea nel mezo, come &longs;i co&longs;tuma, & mol­<lb/>tiplicando il tre intero co'l due denominatore <lb/>della metà, ne viene 6, alquale di &longs;opra &longs;i ag­<lb/>giunge vno, & è vn &longs;e&longs;to. </s> </p> <p type="main"> <s><emph type="italics"/>Che &longs;e come nella terza figura la corda &longs;i allunghe <lb/>rà in O, & &longs;i condarrà intorno ad vn'altra gi­<lb/>rella, il cui centro &longs;ia Q, la qual corda poi &longs;i <lb/>leghi in R alla taglia di &longs;otto; &longs;arà la po&longs;&longs;an­<lb/>za di G vn &longs;ettimo del pe&longs;o. </s> <s>& co&longs;i proceden <lb/>do in in&longs;inito, la proportione della po&longs;&longs;anza al pe<emph.end type="italics"/> <arrow.to.target n="note238"></arrow.to.target><lb/><emph type="italics"/>&longs;o, quanto &longs;i voglia &longs;otto moltiplice ver&longs;o il pe­<lb/>&longs;o &longs;i potrà trouare. </s> <s>Dapoi &longs;i mo&longs;trerà &longs;empre, <lb/>come nelle precedenti, che &longs;e la po&longs;&longs;anza, la­<lb/>quale &longs;o&longs;tiene il pe&longs;o &longs;arà vn quarto, ouero vn <lb/>quinto, ouero in qual &longs;i voglia altro modo &longs;arà <lb/>di&longs;po&longs;ta ver&longs;o il pe&longs;o, che &longs;imilmente cia&longs;cuna <lb/>corda &longs;o&longs;terrà la quarta, ò la quinta, ouero qual <lb/>&longs;i voglia altra parte del pe&longs;o, &longs;i come la i&longs;te&longs;&longs;a <lb/>po&longs;&longs;anza: peroche le corde fanno il mede&longs;imo, <lb/>come &longs;e fo&longs;&longs;ero tante po&longs;&longs;anze: & le girelle co­<lb/>me &longs;e fo&longs;&longs;ero tante leue.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note238"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 8. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s>Sotto moltiplice. </s> <s>Que&longs;to è il primo genere delle <lb/>proportioni, che &longs;i riguardano dal minore al <lb/>maggiore, detto di minore di&longs;uguaglianza, il <lb/>quale &longs;otto di &longs;e tiene a&longs;&longs;ais&longs;ime &longs;petie, & è op­<lb/>po&longs;to come ho ricordato, al moltiplice. </s> <s>Dice <lb/>l'autore: & co&longs;i procedendo in infinito &longs;i potrà <lb/>ritrouare qual &longs;i voglia proportione &longs;otto mol <lb/>tiplice. </s> <s>Percio che la po&longs;&longs;anza è minore del pe <lb/>&longs;o, & però ver&longs;o lui ha proportione &longs;otto mol <lb/>tiplice, come di vno ver&longs;o due, & di due ver­<lb/>&longs;o quattro per darne e&longs;&longs;empio, & co&longs;i de gli al­<lb/>tri numeri tali. </s> </p> <figure></figure> <pb/> <p type="head"> <s>COROLLARIO</s> </p> <p type="main"> <s>Di qui è manife&longs;to, che le girelle della taglia, allaquale è legato <lb/>il pe&longs;o, fanno sì, che il pe&longs;o è &longs;o&longs;tenuto da po&longs;&longs;anza minore, <lb/>di quel che &longs;ia e&longs;&longs;o pe&longs;o, co&longs;a che veramente non fanno le gi­<lb/>relle della taglia di &longs;opra. </s> </p> <p type="main"> <s><emph type="italics"/>Egli nondimeno conuiene &longs;apere, che come &longs;uole &longs;ar&longs;i, la girella della taglia di &longs;otto, <lb/>il cui centro è N, deue e&longs;&longs;ere minore di quella girella, il cui centro è C, & que <lb/>&longs;ta anche minore di quella, che ha il centro in B: & in &longs;omma &longs;e &longs;aranno più gi <lb/>relle nella taglia di &longs;otto legata al pe&longs;o, &longs;empre quella girella deue e&longs;&longs;ere maggiore <lb/>delle altre, che è più vicina al pe&longs;o attaccato: ma al contrario hanno à di&longs;por&longs;i le <lb/>girelle nella taglia di &longs;opra, ilche &longs;i co&longs;tuma di fare, acciò che le corde fra loro non <lb/>&longs;i intrichino; peroche in quanto alle girelle, &longs;iano ò grandi, ò picciole, non importa <lb/>nulla, &longs;eguendone &longs;empre l'i&longs;te&longs;&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Di più è da notare, ilche etiandio dalle co&longs;e dette facilmente appare, che grandi&longs;&longs;ima <lb/>differenza na&longs;ce trala po&longs;&longs;anza, & il pe&longs;o dal legare la corda ouero in R della ta <lb/>glia di &longs;otto, ouero in S, percioche &longs;e &longs;i legherà in S, la po&longs;&longs;anza di G &longs;arà vn <lb/>&longs;e&longs;to del pe&longs;o; ma &longs;e in R vn &longs;ettimo, co&longs;a che non accade alla taglia di &longs;opra: <lb/>percioche leghi&longs;i la corda, come nella precedente figura, ouero in T, ouero in O, <lb/>&longs;empre la po&longs;&longs;anza di G &longs;arà vn &longs;e&longs;to di e&longs;&longs;o pe&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Dopo que&longs;te co&longs;e egli è da con&longs;iderare in che modo la forza moua il pe&longs;o, & di più lo <lb/>&longs;patio, & il tempo della po&longs;&longs;anza, che moue, & del pe&longs;o che è mo&longs;&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s>“Di piu egli è da notare ilche etiandio è manife&longs;to dalle co&longs;e dette &c. </s> <s>Qui potreb­<lb/>be for&longs;e ad alcuno parere difficile in che modo po&longs;&longs;a e&longs;&longs;ere, che dal legare la cor­<lb/>da in R, ouero in S, come &longs;i vede in que&longs;ta figura, na&longs;ca tanta differenza. </s> <s>Onde <lb/>noti&longs;i che legando la corda in S, la girella Q re&longs;ta del tutto inutile, & è come <lb/>&longs;e ella non vifo&longs;&longs;e; & la corda per non e&longs;&longs;ere attaccata in R alla taglia di &longs;otto, <lb/>ma in S fuori non &longs;o&longs;tiene la taglia, talche la forza di G viene ad e&longs;&longs;ere &longs;olamen <lb/>te vn &longs;e&longs;to del pe&longs;o. </s> <s>&longs;oggiunge poi ilche non auiene alla taglia di &longs;opra.” Doue <lb/>auerta&longs;i che mentre &longs;i ha tenuto propo&longs;ito delle lettere S & R, ha bi&longs;ognato guar <lb/>dare nella qui &longs;opra&longs;critta figura, ma in parlando di TO, egli è me&longs;tieri per in­<lb/>tendere que&longs;to loco mirare nella figura precedente, che è la &longs;econda della nona <lb/>propo&longs;itione, peroche iui&longs;ono le lettere TO. </s> <s>La ragione per la quale non na&longs;ca <lb/>differenza nella po&longs;&longs;anza à legare la corda in T ouero in O, ma &longs;ia tutto vno, <lb/>è che la taglia di &longs;opra &longs;ta &longs;empre ferma, per modo, che non importa nulla il le­<lb/>gare la corda in O nella taglia di &longs;opra, ouero in T fuori di e&longs;&longs;a, poiche am­<lb/>bidue i luoghi &longs;ono immobili, & iui la corda &longs;ta ferma. </s> <s>Lequali tutte co&longs;e l'auto <lb/>re hà toccato breuis&longs;imamente per e&longs;&longs;ere que&longs;to trattato della taglia lungo, la­<lb/>&longs;ciando al lettore ancora qualche co&longs;a da &longs;peculare per &longs;e mede&longs;imo. </s> </p> <pb pagenum="71"/> <p type="head"> <s>PROPOSITIONE X.</s> </p> <p type="main"> <s>Se la corda &longs;arà inuolta intorno alla girella della taglia applcca­<lb/>ta di &longs;opra, all'vno de'capi, dellaqual corda &longs;ia attaccato il pe <lb/>&longs;o, & all'altro po&longs;ta la po&longs;&longs;anza, che moue. </s> <s>La detta po&longs;&longs;anza mo <lb/>uerà con la leua &longs;empre egualmente di&longs;tante dall'orizonte. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A. </s> <s>&longs;ia la girella della taglia appiccata di &longs;opra, che habbia il centro K. </s> <s><lb/>Sia dapoi la corda HB CDEF legata al pe&longs;o A in H, & &longs;ia inuolta d'intor <lb/>no alla girella; & &longs;ia la taglia per modo appiccata <lb/>in L, che non habbia alcun altro mouimento &longs;uor <lb/>che il volgimento libero della girella d'intorno al <lb/>&longs;uo a&longs;&longs;etto, & &longs;ia la po&longs;&longs;anza in F che moua il <lb/>pe&longs;o A. </s> <s>Dico, che la po&longs;&longs;anza di F mouerà <lb/>&longs;empre il pe&longs;o A con la leua egualmente di&longs;tan­<lb/>te dall'orizonte. </s> <s>&longs;ia tirata la linea BKE egual­<lb/>mente di&longs;tante dall'orizonte, & &longs;iano i punti BE<emph.end type="italics"/> <arrow.to.target n="note239"></arrow.to.target><lb/><emph type="italics"/>doue le corde BH & EF toccano il cerchio: <lb/>&longs;arà BKE la leua, il &longs;o&longs;tegno dellaquale è nel <lb/>&longs;uo mezo, che è K, come di &longs;opra è detto. </s> <s>Men­<lb/>tre che dunque la forza di F inchina al ba&longs;&longs;o ver <lb/>&longs;o M, la leua EB &longs;i mouerà, mouendo&longs;i tut­<lb/>ta la girella, cioè volgendo&longs;i attorno. </s> <s>Mentre <lb/>che dunque F &longs;ta in M &longs;ia il punto E della <lb/>leua mo&longs;&longs;o fin ad I, & il B &longs;in'al C, di mo­<lb/>do, che la leua &longs;ia in CI. </s> <s>Dapoi &longs;i faccia la li <lb/>nea NM eguale ad e&longs;&longs;a FE: & quando il <lb/>punto E, &longs;arà in I all'hora il punto della cor <lb/>da, ilquale era in E &longs;arà in N, & quello,<emph.end type="italics"/><lb/> <arrow.to.target n="fig96"></arrow.to.target><lb/><emph type="italics"/>che era in B &longs;arà in C di modo, che tirata la linea CI pa&longs;&longs;erà per lo centro <lb/>K. </s> <s>Hormentre il B &longs;ta in C &longs;ia il punto H in G, & &longs;arà BH al CBG <lb/>eguale, e&longs;&longs;endo la mede&longs;ima corda. </s> <s>& percioche mentre EF inchina in MN <lb/>rimane pur &longs;empre EFM à piombo dell'orizonte, & tocca il cerchio nel punto <lb/>E di modo, che la linea tirata dal punto E per lo centro K &longs;ia &longs;empre egualmen <lb/>te di&longs;tante dall'orizonte, ilche mede&longs;imamente auiene alla corda BG & al pun- <pb/>to B. </s> <s>Mentre dunque il cerchio, ouero la girella &longs;i volge intorno, &longs;empre &longs;i mo­<lb/>ue la leua EB, & &longs;em­<lb/>pre ancora rimane vn'al­<lb/>tra leua in EB, e&longs;&longs;endo <lb/>che per natura di e&longs;&longs;a gi­<lb/>rella, nellaquale &longs;empre, <lb/>mentre &longs;i moue, re&longs;ti il <lb/>diametro da B in E, <lb/>(ilquale è in loco di le­<lb/>ua) auuiene che parten <lb/>do&longs;ene vna, &longs;ucceda <lb/>l'altra &longs;empre, durando <lb/>però cotale aggiramen­<lb/>to; & co&longs;i accade, che <lb/>la po&longs;&longs;anza moua il pe <lb/>&longs;o &longs;empre con la leua <lb/>EB egualmente di&longs;tan <lb/>te dall'orizonte, ilche <lb/>bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note239"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig96"></figure> <figure></figure> <p type="main"> <s>Po&longs;te le co&longs;e i&longs;te&longs;&longs;e, lo &longs;patio della po&longs;&longs;anza, che moue il pe&longs;o, è <lb/>eguale allo &longs;patio dello i&longs;te&longs;&longs;o pe&longs;o, che è mo&longs;&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche egli è &longs;tato dimo&longs;trato, che mentre F &longs;tà in M, il pe&longs;o A, cioè il punto <lb/>H è in G: & concio&longs;ia che la corda HBCDEF &longs;ia eguale alla GBCDEN <lb/>FM per e&longs;&longs;ere la corda i&longs;te&longs;&longs;a: leuata via dunque la commune GBCDENF <lb/>&longs;arà la HG alla FM eguale, & &longs;imilmente &longs;i mo&longs;trerà la di&longs;ce&longs;a di F e&longs;&longs;ere <lb/>&longs;empre eguale alla &longs;alita di H. </s> <s>Adunque lo &longs;patio della po&longs;&longs;anza è eguale allo <lb/>&longs;patio del pe&longs;o. </s> <s>che era da dimo&longs;trare.<emph.end type="italics"/></s> </p> <p type="main"> <s>Oltre à ciò la po&longs;&longs;anza moue il pe&longs;o i&longs;te&longs;&longs;o per i&longs;patio eguale in <lb/>tempo eguale, tanto con la corda inuolta intorno alla girella <lb/>della taglia appiccata di &longs;opra, quanto &longs;enza taglia, pur che li <lb/>mouimenti di e&longs;&longs;a po&longs;&longs;anza in velocità &longs;iano eguali. </s> </p> <pb pagenum="72"/> <p type="main"> <s><emph type="italics"/>Stando le co&longs;e i&longs;te&longs;&longs;e, &longs;ia vn'altro pe&longs;o P eguale al pe&longs;o A, alquale &longs;ia legata la cor <lb/>da TQ à piombo dell'orizonte: & &longs;ia TQ eguale ad e&longs;&longs;a HB: & muoua <lb/>la po&longs;&longs;anza di Q il <lb/>pe&longs;o P all'insù ad <lb/>angoli retti all'orizon <lb/>te, come &longs;i moue il pe <lb/>&longs;o A. </s> <s>Dico, che per <lb/>eguale &longs;patio, & in <lb/>vno i&longs;te&longs;&longs;o tempo la <lb/>po&longs;&longs;anza di <expan abbr="q.">que</expan> mo­<lb/>ue il pe&longs;o P, & la <lb/>po&longs;&longs;anza di<emph.end type="italics"/> F <emph type="italics"/>il pe­<lb/>&longs;o A: ilche è il me­<lb/>de&longs;imo, come &longs;e l'i­<lb/>&longs;te&longs;&longs;o pe&longs;o fo&longs;&longs;e mo&longs;­<lb/>&longs;o in tempo eguale, <lb/>&longs;econdo che habbia­<lb/>mo propo&longs;to. </s> <s>Sia <lb/>allungata la EF in <lb/>S, & la TQ in R, <lb/>& &longs;iano le QRFS <lb/>fatte eguali non &longs;olo <lb/>fra &longs;e, ma etiandio <lb/>ad e&longs;&longs;a BH. </s> <s>Hor <lb/>concio&longs;ia che le TQ <lb/>QR &longs;iano eguali ad <lb/>e&longs;&longs;e HB FS, & <lb/>la &longs;orza di Q mo­<lb/>ua il pe&longs;o P per <lb/>la linea retta TQ <lb/>R: & dall'altro<emph.end type="italics"/><lb/> <arrow.to.target n="fig97"></arrow.to.target><lb/><emph type="italics"/>canto la forza di F moua A per la retta HB, & le velocità de i mouimenti <lb/>dell'una, & l'altra po&longs;&longs;anza &longs;iano eguali, all'hor che nell'i&longs;te&longs;&longs;o tempo la po&longs;&longs;anza di <lb/>Q &longs;arà in R, & la po&longs;&longs;anza di F &longs;arà in S, e&longs;&longs;endo gli &longs;patij eguali: & men <lb/>tre la po&longs;&longs;anza di Q è in R, il pe&longs;o P, cioè il punto T &longs;arà in Q, per e&longs;&longs;e­<lb/>rela TQ eguale ad e&longs;&longs;a QR, & mentre che la po&longs;&longs;anza di F &longs;ta in S, il pe­<lb/>&longs;o A, cioè il punto H &longs;arà in B; ma lo &longs;patio TQ è eguale allo &longs;patio HB: <lb/>adunque le po&longs;&longs;anze di FQ mo&longs;&longs;e egualmente moueranno i pe&longs;i PA eguali <lb/>per eguali &longs;patij in tempo eguale. </s> <s>che era da mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig97"></figure> <p type="head"> <s>PROPOSITIONE XI.</s> </p> <p type="main"> <s>Se la corda &longs;arà inuolta intorno alla girella della taglia legata al <lb/>pe&longs;o, laqual corda con vno de' &longs;uoi capi &longs;ia legata in qualche <lb/>luogo, & con l'altro pre&longs;a dalla po&longs;&longs;anza che moue il pe&longs;o; La <lb/>po&longs;&longs;anza mouerà &longs;empre con la leua egualmente di&longs;tante dal <lb/>l'orizonte. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A: &longs;ia la girella CED <lb/>della taglia legata al pe&longs;o A, <lb/>da KH, & &longs;ia KH ad ango­<lb/>li retti dell'orizonte, di modo <lb/>the il pe&longs;o &longs;egua &longs;empre il mo­<lb/>uimento della taglia, &longs;ia pur fat­<lb/>to all'insù, ouero all'ingiù, & <lb/>&longs;ia il centro della girella K, & <lb/>la corda inuolta intorno alla gi­<lb/>rella &longs;ia BCDEF, la quale <lb/>&longs;ia legata in B, di modo che <lb/>&longs;tia immobile in B<gap/> & &longs;ia in F <lb/>la po&longs;&longs;anza, che moue il pe&longs;o A. </s> <s><lb/>Dico che la po&longs;&longs;anza di F mo­<lb/>ue &longs;empre il pe&longs;o A con la le­<lb/>ua egualmente di&longs;tante dall'ori­<lb/>zonte. </s> <s>Siano BC EF egual <lb/>mente di&longs;tanti sì fra loro, come <lb/>ad e&longs;&longs;a KH, & à piombo al-<emph.end type="italics"/><lb/> <arrow.to.target n="note240"></arrow.to.target> <emph type="italics"/>l'orizonte della i&longs;te&longs;&longs;a KH, & <lb/>toccanti il cerchio CED ne i <lb/>punti EC, & &longs;ia congiunta la <lb/>EC laquale pa&longs;&longs;erà per lo cen <lb/>tro K, & &longs;arà egualmente di <lb/>&longs;tante dall'orizonte, &longs;i come pri­<lb/>ma è detto. </s> <s>Hor percioche la <lb/>girella CED &longs;i volge d'intor­<lb/>no K &longs;uo centro, però mentre <lb/>la forza di F tira sù il punto E <lb/>dourebbe di&longs;cendere il punto C <lb/>& tirare in giù B: mala cor­<lb/>da po&longs;ta in B è immobile, on­<lb/>de BC non può di&longs;cendere. </s> <s><lb/>Per laqual co&longs;a mentre la po&longs;-<emph.end type="italics"/><lb/> <arrow.to.target n="fig98"></arrow.to.target> <pb pagenum="73"/><emph type="italics"/>&longs;anza di F tira sùlo E, tutta la girella &longs;i mouerà in sù, & per con&longs;equenza tut­<lb/>ta la taglia, & il pe&longs;o; & EKC &longs;arà come leua, il cui &longs;o&longs;tegno &longs;arà C: pero-<emph.end type="italics"/> <arrow.to.target n="note241"></arrow.to.target><lb/><emph type="italics"/>che il punto C per cau&longs;a di BC qua&longs;i è immobile, ma la po&longs;&longs;anza che moue la <lb/>leua è in F con la corda EF, & il pe&longs;o &longs;ta appiccato in K. </s> <s>Che &longs;e il punto <lb/>C fo&longs;&longs;e del tutto immobile, & &longs;i moua la leua EC in NC, & &longs;i diuida NC <lb/>in due parti eguali in L: &longs;aranno CL LN eguali ad e&longs;&longs;e CK KE. </s> <s>Per la <lb/>qual co&longs;a &longs;e la leua EC fo&longs;&longs;e in CN, il punto K &longs;arebbe in L: & &longs;e &longs;i con <lb/>duce&longs;&longs;e la linea LM à piombo dell'orizonte, laquale &longs;ia anche eguale alla KH, <lb/>&longs;arebbe il pe&longs;o A, cioè il punto H in M. </s> <s>Ma percioche la po&longs;&longs;anza di F men <lb/>tre và in &longs;u&longs;o mouendo la girella &longs;empre &longs;i moue &longs;opra la linea retta EFG, laquale <lb/>è anco egualmente di&longs;tante &longs;empre da BC, &longs;arà nece&longs;&longs;ario, che la girella della ta­<lb/>glia &longs;empre &longs;i trouitra le linee EG BC, & il centro K &longs;tando nel mezo, &longs;i mo­<lb/>uerà &longs;empre &longs;opra la linea retta HKT. </s> <s>Sia condotta adunque per L la linea <lb/>PT LQ egualmente di&longs;tante sì dall orizonte, come dalla EC, laquale &longs;eghi la <lb/>HK allu<gap/>gatain T, & co'l centro T, & lo &longs;patio TQ &longs;i formi il cerchio QR <lb/>PS, ilquale &longs;arà eguale al cerchio CED; & li punti PQ toccheranno le cor­<lb/>de FE BC ne i punti <expan abbr="Pq.">Pque</expan> Peroche il rettangolo PECQ & la PT & la <lb/>TQ &longs;ono eguali ad e&longs;&longs;e EK KC. </s> <s>Dapoi per T &longs;ia tirato RTS diametro <lb/>del cerchio PQS egualmente di&longs;tante ad e&longs;&longs;a NC, & &longs;ia fatta TO eguale<emph.end type="italics"/> <arrow.to.target n="note242"></arrow.to.target><lb/><emph type="italics"/>alla KH. </s> <s>Hor mentre il centro K &longs;arà mo&longs;&longs;o fin alla linea PQ all'hora il cen­<lb/>tro K &longs;arà in T. </s> <s>Maegliè &longs;tato dimo&longs;trato, che il centro della girella &longs;i moue <lb/>&longs;empre per la linearetta HT. </s> <s>Onde accioche il centro K &longs;ia nella linea PQ egual­<lb/>mente di&longs;tante ad e&longs;&longs;a EC, egli è nece&longs;&longs;ario, che e&longs;&longs;o &longs;ia in T: & accioche an­<lb/>chora la leua EC &longs;i alzinell'angolo ECN egli è nece&longs;&longs;ario, che &longs;ia in RS & <lb/>non in CN percioche l'angolo RSE all'angolo NCE è eguale & co&longs;i il &longs;o-<emph.end type="italics"/> <arrow.to.target n="note243"></arrow.to.target><lb/><emph type="italics"/>&longs;tegno C non è del tutto immobile, mouendo&longs;i tutta la girella all'insù, & tutta <lb/>mutt'il luogo: nondimeno il C ha ragione di &longs;o&longs;tegno, peroche meno &longs;i moue C <lb/>di quel che fà K & E, percioche &longs;i moue il punto E fin ad R, & il K fin al T, <lb/>ma il punto C fin ad S &longs;olamente. </s> <s>Per laqual co&longs;a mentre il centro K &longs;i troua <lb/>in T, il &longs;ito della girella &longs;arà QRPS: & il pe&longs;o A, cioè il punto H &longs;arà <lb/>in O, e&longs;&longs;endo TO eguale à KH; ma il &longs;ito di EC, cioè della leua mo&longs;&longs;a, &longs;arà <lb/>RS: & la po&longs;&longs;anza di F &longs;arà mo&longs;&longs;a in &longs;u&longs;o per la rettalinea EFG: ma nel­<lb/><gap/>i&longs;te&longs;&longs;o tempo, che K &longs;arà in T, &longs;ia la po&longs;&longs;anza in G; & mentre la leua EC in <lb/>que&longs;to modo &longs;i moue, rimangono pur &longs;empre GPBQ fraloro egua mente di­<lb/>&longs;tanti, & à piombo dell'orizonte, talche doue toccano la girella, come ne' punti <lb/>PQ, &longs;empre la linea PQ &longs;arà il diametro della girella & come leua egualmen­<lb/>te di&longs;tante dall'orizonte. </s> <s>Mentre dunque la girella &longs;i moue, & và attorno, &longs;em <lb/>pre anche &longs;i moue la leua EC, & &longs;empre rimane vn'altra leua nellagirella egual <lb/>mente di&longs;tante dall'orizonte, come PQ, per modo, che la po&longs;&longs;anza di F <gap/>ona <lb/>il pe&longs;o, &longs;tando la leua egualmente di&longs;tante all'orizonte, il cui &longs;o&longs;tegno &longs;acà &longs;e<gap/>pre <lb/>nellalinea CB, & il pe&longs;o nel mezo della leua appiccato: & la po&longs;&longs;anza nella li­<lb/>nea EG, che era da mo&longs;trare.<emph.end type="italics"/></s> </p> <pb/> <p type="margin"> <s><margin.target id="note240"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note241"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note242"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 34. <emph type="italics"/>del pri<gap/>o.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note243"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/><gap/><emph.end type="italics"/></s> </p> <figure id="fig98"></figure> <p type="main"> <s>Stando le co&longs;e i&longs;te&longs;&longs;e. </s> <s>Lo &longs;patio della po&longs;&longs;anza, che moue il pe­<lb/>&longs;o è il doppio dello &longs;patio dell'i&longs;te&longs;&longs;o pe&longs;o mo&longs;&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>E&longs;&longs;endo &longs;tato dimo&longs;trato, che mentre il K &longs;tà nel T, il pe&longs;o A cioè il punto H <lb/>e&longs;&longs;ere in O: & nell'i&longs;te&longs;&longs;o tempo ancora la po&longs;&longs;anza di F e&longs;&longs;ere in G: & per­<lb/>cioche la corda BCDEF eguale è alla corda EQSPG, peroche è la mede&longs;ima <lb/>corda: & la corda che è inuolta intorno al mezo cerchio CDE eguale è alla cor­<lb/>da, che &longs;ta d'intorno al mezo cerchio QSP: tolti via dunque li due pezzi di cor <lb/>da communi BQ, & FP: &longs;arà il re&longs;lante della corda FG eguale ad e&longs;&longs;i due <lb/>pezzi di corda rima&longs;i CQ & EP in&longs;ieme pre&longs;i. </s> <s>Ma EP eguale è al TK, <lb/>& il CQ &longs;arà anche eguale ad e&longs;&longs;o TK, peroche &longs;ono PK & TC parallelo­<lb/>grammi rettangoli. </s> <s>Per laqual co&longs;a le linee EPCQ in&longs;ieme &longs;ono due volte tan <lb/>to, quanto è TK. </s> <s>Adunque la corda FC &longs;arà due volte tanto quanto la TK. <lb/>& percioche la KH è eguale alla TO, leuando via la corda commune KO &longs;a <lb/>rà la KT eguale ad e&longs;&longs;a KO. </s> <s>Per laqual co&longs;a la corda FG &longs;arà due volte tan­<lb/>to quanto e&longs;&longs;a HO: cioè lo &longs;patio d'lla po&longs;&longs;anza due volte tanto quanto lo &longs;pa­<lb/>tio del pe&longs;o, che era da mo&longs;tra<gap/>e.<emph.end type="italics"/></s> </p> <p type="main"> <s>“Parallelogrammi rettangoli. </s> <s>Vuol dire &longs;igure di linee egualmente diftanti fra loro, <lb/>lequali formino angoli retti à differenza di altre figure, che &longs;e ben &longs;ono di linee <lb/>egualmente di&longs;tanti, non formano tuttauia angoli retti. </s> </p> <p type="main"> <s>Dapoi la po&longs;&longs;anza mouerà il pe&longs;o i&longs;te&longs;&longs;o in tempo eguale per la <lb/>metà dello &longs;patio, con la corda inuolta d'intorno alla girella <lb/>della taglia legata al pe&longs;o, che &longs;enza taglia; pur che le veloci­<lb/>tà de' mouimenti di e&longs;&longs;a po&longs;&longs;anza &longs;iano eguali. </s> </p> <pb pagenum="74"/> <p type="main"> <s><emph type="italics"/>Peroche &longs;ia, &longs;tando le co&longs;e i&longs;te&longs;&longs;e, vn'altro pe&longs;o V eguale al pe&longs;o A al quale &longs;ia <lb/>legata la corda <36>X & &longs;ia in X la po&longs;&longs;anza, che moue il pe&longs;o V, Dico, &longs;e le ve <lb/>locità de' mouimenti dell'vna, & l'altra po&longs;&longs;anza &longs;aranno eguali, che la po&longs;&longs;anza<emph.end type="italics"/><lb/> <arrow.to.target n="fig99"></arrow.to.target><lb/><emph type="italics"/>di<emph.end type="italics"/> F <emph type="italics"/>mouerà il pe&longs;o A nell'i&longs;te&longs;&longs;o tempo per la metà dello &longs;patio, per lo quale <lb/>il pe&longs;o V &longs;arà mo&longs;&longs;o dalla po&longs;&longs;anza di X, che è il mede&longs;imo, come &longs;el'i&longs;te&longs;&longs;o pe­<lb/>&longs;o in tempo eguale fo&longs;&longs;e mo&longs;&longs;o. </s> <s>Moua la po&longs;&longs;anza di X il pe&longs;o V, & la po&longs;&longs;an­<lb/>za peruenga in <foreign lang="greek">*u</foreign>; & &longs;ia X<foreign lang="greek">*u</foreign> eguale ad e&longs;&longs;a FG: & &longs;i faccia <foreign lang="greek">*u</foreign>Z eguale <lb/>à X<36>, talche quando la po&longs;&longs;anza di X &longs;arà in <foreign lang="greek">*u</foreign>, &longs;ia il pe&longs;o V cioè il punto <36><emph.end type="italics"/> <pb/><emph type="italics"/>in Z; ma <36>Z è eguale ad FG, e&longs;&longs;endo eguale ad X<foreign lang="greek">*u</foreign>: dunque <36>Z &longs;arà due <lb/>volte tanto, quanto OH. </s> <s>Per laqual co&longs;a mentre le po&longs;&longs;anze &longs;aranno in G<foreign lang="greek">*u</foreign>, i <lb/>pe&longs;i AV &longs;aranno in OZ. </s> <s>Hor nell'i&longs;te&longs;&longs;o tempo &longs;aranuo le po&longs;&longs;anze in G<foreign lang="greek">*u</foreign>, <lb/>peroche le vetocità de mouimenti &longs;ono eguali: onde la forza di F mouerà il pe­<lb/>&longs;o A nel mede&longs;imo tempo per la metà di quello &longs;patio, per loquale il pe&longs;o V &longs;a<emph.end type="italics"/><lb/> <arrow.to.target n="fig100"></arrow.to.target><lb/><emph type="italics"/>rà mo&longs;&longs;o dalla po&longs;&longs;anza di X: & li pe&longs;i &longs;ono eguali, adunque la po&longs;&longs;anza moue­<lb/>rà il pe&longs;o i&longs;l e&longs;&longs;o in tempo eguale per la metà dello &longs;patio, con la corda, & la taglia <lb/>legata in que&longs;to modo al pe&longs;o, che &longs;enza taglia; purche le velocità della po&longs;&longs;anza <lb/>de'mouimenti &longs;iano eguali, che era da mo&longs;trar&longs;i.<emph.end type="italics"/></s> </p> <pb pagenum="75"/> <figure id="fig99"></figure> <figure id="fig100"></figure> <p type="head"> <s>PROPOSITIONE XII.</s> </p> <p type="main"> <s>Se la corda &longs;arà riuolta d'intorno à più girelle, legando l'vno de' <lb/>capi &longs;uoi in qualche loco, & l'altro &longs;ia tenuto dalla po&longs;&longs;anza, <lb/>che moue il pe&longs;o: La po&longs;&longs;anza mouerà con le leue &longs;empre <lb/>egualmente di&longs;tanti dall'orizonte. </s> </p> <p type="main"> <s><emph type="italics"/>Siail pe&longs;o A. </s> <s>&longs;ia la girella CED della <lb/>taglia legata al pe&longs;o da KS ad angoli ret <lb/>ti all'orizonte; di modo, che il pe&longs;o &longs;egua <lb/>&longs;empre il &longs;uo mouimento ò &longs;u&longs;o, ò giu&longs;o, <lb/>che &longs;ia fatto. </s> <s>Sia dapoi la girella intorno <lb/>al centro L della taglia appiccata di &longs;opra; <lb/>& &longs;iala corda BCDEHMNO riuol­<lb/>ta d'intorno alle girelle, laquale &longs;ia legata <lb/>in B; & &longs;ia in O la forza mouente il <lb/>pe&longs;o A, mouendo&longs;i al ba&longs;&longs;o per OP. </s> <s>Di­<lb/>co che la po&longs;&longs;anza di O mouerà &longs;empre il <lb/>pe&longs;o A con le leue &longs;empre egualmente <lb/>di&longs;tanti dall'orizonte. </s> <s>&longs;ia tirata la linea <lb/>NH per lo centro L egualmente di&longs;tan­<lb/>te dall'orizonte, che &longs;arà la leua della girel-<emph.end type="italics"/> <arrow.to.target n="note244"></arrow.to.target><lb/><emph type="italics"/>la, il cui centro è L: &longs;ia tirata da poi la <lb/>EC per lo centro K, &longs;imilmente di&longs;tan-<emph.end type="italics"/> <arrow.to.target n="note245"></arrow.to.target><lb/><emph type="italics"/>te egualmente dall'orizonte, la quale &longs;arà <lb/>anche la leua della girella, il cui centro è<emph.end type="italics"/> <arrow.to.target n="note246"></arrow.to.target><lb/><emph type="italics"/>K. </s> <s>Moua&longs;i la po&longs;&longs;anza di O in giu&longs;o, la <lb/>quale mentre in giu&longs;o &longs;i moue, mouerà la <lb/>leua NH, & mentre la leua &longs;i moue, la <lb/>N &longs;i mouerà in giu&longs;o, & la H in &longs;u&longs;o, <lb/>come è detto di &longs;opra. </s> <s>Mamentre la H <lb/>&longs;i moue in &longs;u&longs;o, moue etiandio in &longs;u&longs;o la E, <lb/>& la leua EC, il cui &longs;o&longs;tegno è C, ma <lb/>il &longs;o&longs;tegno C non puote mouere in giu&longs;o <lb/>il B; però la girella il cui centro è K mo <lb/>uera&longs;&longs;i in &longs;u&longs;o, & per con&longs;equenza la ta­<lb/>glia, & il pe&longs;o A, come nella preceden­<lb/>te è stato detto. </s> <s>& perche per la mede&longs;i­<lb/>ma cau&longs;a, che è stata a&longs;&longs;egnata nelle pre­<lb/>cedenti, rimangono &longs;empre le leue egual­<lb/>mente distanti dall'orizonte in HN, &<emph.end type="italics"/><lb/> <arrow.to.target n="fig101"></arrow.to.target><lb/><emph type="italics"/>in EC, la po&longs;&longs;anza dun­<lb/>que mouente il pe&longs;o A <lb/>lo mouerà &longs;empre &longs;tando <lb/>le leue egualmente distan­<lb/>ti dall'orizonte; che erada <lb/>mo&longs;trar&longs;i.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note244"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>&<emph.end type="italics"/> 10. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note245"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note246"></margin.target><emph type="italics"/>Par la<emph.end type="italics"/> 10. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig101"></figure> <p type="main"> <s><emph type="italics"/>Et &longs;e la corda &longs;arà riuolta d'in <lb/>torno à più girelle; &longs;imil­<lb/>mente &longs;i dimo&longs;trerà la po&longs;­<lb/>&longs;anza mouere il pe&longs;o con <lb/>le leue &longs;empre egualmente <lb/>di&longs;tanti dall'orizonte: & <lb/>le leue delle girelle della ta <lb/>glia di &longs;opra &longs;empre e&longs;&longs;e­<lb/>re come HN, i &longs;o&longs;tegni <lb/>delle quali &longs;aranno &longs;empre <lb/>nel mezo: ma le leue delle <lb/>girelle della taglia di &longs;otto <lb/>&longs;empre e&longs;&longs;ere, come EC; <lb/>li cui &longs;o&longs;tegni &longs;aranno nel­<lb/>le &longs;tremità delle leue.<emph.end type="italics"/></s> </p> <p type="main"> <s>Stando le co&longs;e i&longs;te&longs;&longs;e, <lb/>lo &longs;patio della po&longs;­<lb/>&longs;anza, è il doppio <lb/>dello &longs;patio del pe­<lb/>&longs;o. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia mo&longs;&longs;o il centro K fin al centro R; & &longs;ia la girella FTG: poi &longs;ia per lo cen­<lb/>tro R condotta la linea GF egualmente di&longs;tante da e&longs;&longs;a EC: le corde EH <lb/>CB toccheranno la girella ne i punti GF. </s> <s>Faccia&longs;i alla fine RQ eguale à <lb/>KS. </s> <s>Mentre dunque K &longs;arà in R, il pe&longs;o A, cioè il punto S &longs;arà in Q,<emph.end type="italics"/> <pb pagenum="76"/><emph type="italics"/>& mentre il centro della girella è in R, &longs;ia la po&longs;&longs;anza di O mo&longs;&longs;a in P. </s> <s>& <lb/>percioche la corda BCDEHMNO eguale è alla corda BFTGHMNP <lb/>per e&longs;&longs;er la corda iste&longs;&longs;a, & FTG è eguale à CDE; leuate via dunque le com­<lb/>muni BF & GHMNO, &longs;arà la re&longs;tante OP eguale ad e&longs;&longs;e FC EG pre­<lb/>&longs;e in&longs;ieme: & per con&longs;equenza due volte tanto, quanto è KR, & QS. </s> <s>& e&longs;­<lb/>&longs;endo OP lo &longs;patio della po&longs;&longs;anza mo&longs;&longs;a, & SQ lo &longs;patio del pe&longs;o mo&longs;&longs;o; <lb/>&longs;arà lo &longs;patio della po&longs;&longs;anza due volte tanto quanto lo &longs;patio del pe&longs;o. </s> <s>che era <lb/>da mostrar&longs;i.<emph.end type="italics"/></s> </p> <p type="main"> <s>Oltre à ciò la po&longs;&longs;anza mouerà il pe&longs;o i&longs;te&longs;&longs;o in tempo eguale <lb/>per la metà dello &longs;patio, con vna corda riuolta d'intorno à <lb/>due girelle, l'una delle quali &longs;ia della taglia di &longs;opra, & l'altra <lb/>&longs;ia della taglia legata al pe&longs;o; che &longs;enza taglie: pur che i mo­<lb/>uimenti di e&longs;&longs;a po&longs;&longs;anza &longs;iano egualmente veloci. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche &longs;tando le co­<lb/>&longs;e i&longs;te&longs;&longs;e, &longs;ia il pe&longs;o <lb/>V eguale ad e&longs;&longs;o A, <lb/>alquale &longs;ia legata la <lb/>corda X<36>; & &longs;ia <lb/>la po&longs;&longs;anza in X che <lb/>moue il pe&longs;o V; la <lb/>quale mentre moue <lb/>il pe&longs;o, peruenga in <lb/><foreign lang="greek">*u</foreign>: & &longs;iano fatte <lb/>X<foreign lang="greek">*u</foreign> Z<36> eguali ad <lb/>e&longs;&longs;a OP; &longs;arà Z<36><lb/>due volte tanto qua­<lb/>to QS. & &longs;e le <lb/>velocità de' moui­<lb/>menti dell'vna, & <lb/>l'altra po&longs;&longs;anza &longs;a­<lb/>ranno eguali; egli è <lb/>manife&longs;to, che il pe­<lb/>&longs;o V trapa&longs;&longs;a due <lb/>volte tanto &longs;patio <lb/>nell'i&longs;te&longs;&longs;o tempo, di <lb/>quel che trapa&longs;&longs;i il <lb/>pe&longs;o A: percioche <lb/>nel tempo mede&longs;imo <lb/>la po&longs;&longs;anza di X per­<lb/>uiene ad <foreign lang="greek">*u</foreign>, & la <lb/>po&longs;&longs;anza di O à P; <lb/>& li pe&longs;i &longs;imilmen­<lb/>te in <expan abbr="Zq.">Zque</expan> che era <lb/>da mo&longs;trar&longs;i.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>PROPOSITIONE XIII.</s> </p> <p type="main"> <s>Riuolgendo la corda d'intorno à due girelle di due taglie, I'vna <lb/>dellequali &longs;ia di &longs;opra, & l'altra di &longs;otto, & legata al pe&longs;o; e&longs;­<lb/>&longs;endo anche l'vno de' capi di detta corda legato alla taglia di <lb/>&longs;otto, & l'altro tenuto dalla po&longs;&longs;anza che moue; &longs;arà lo &longs;patio <lb/>cor&longs;o della po&longs;&longs;anza, che tira, tre volte tanto quanto lo &longs;pati <lb/>del pe&longs;o mo&longs;&longs;o. </s> </p> <pb pagenum="77"/> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A; &longs;ia BCD la girel­<lb/>la della taglia legata al pe&longs;o A, <lb/>attaccato da EQ, & &longs;ia E il <lb/>centro della girella; &longs;ia dapoi F <lb/>GH la girella della taglia appic­<lb/>cata di &longs;opra, il cui centro K; & <lb/>&longs;ia la corda LFGHDBCM ri­<lb/>uolta intorno à tutte le girelle, & <lb/>legata alla taglia di &longs;otto in L: <lb/>& &longs;ia in M la po&longs;&longs;anza, che <lb/>moue. </s> <s>Dico lo &longs;patio cor&longs;o dalla <lb/>po&longs;&longs;anza di M, mentre moue il <lb/>pe&longs;o, e&longs;&longs;ere triplo dello &longs;patio del <lb/>pe&longs;o mo&longs;&longs;o A. </s> <s>Moua&longs;i la po&longs;&longs;an <lb/>za di M fin ad N; & il centro <lb/>E &longs;ia mo&longs;&longs;o fin ad O; & L fin <lb/>à P; & il pe&longs;o A, cioè il pun­<lb/>to Q fin ad R; & la girella <lb/>mo&longs;&longs;a &longs;ia TSV. </s> <s>Siano condot­<lb/>te per EO le linee ST BD <lb/>egualmente di&longs;tanti dall'orizonte, <lb/>lequali &longs;aranno anche tra loro e­<lb/>gualmente di&longs;tanti. </s> <s>Ma percio­<lb/>che mentre E &longs;ta in O, il pun­<lb/>to Q &longs;ta in R; &longs;arà EQ egua <lb/>le ad OR, & EO ade&longs;&longs;o QR <lb/>eguale; &longs;imilmente LQ &longs;arà <lb/>eguale à PR, & LP ad e&longs;&longs;o <lb/>QR eguale. </s> <s>Adunque le tre <lb/>QR EO LP fra loro &longs;aranno <lb/>eguali; à cui &longs;ono etiandio eguali <lb/>BS DT. </s> <s>Et percioche la corda <lb/>LFGHDCBM è eguale alla <lb/>corda PFGHTVSN e&longs;&longs;en­<lb/>do vna corda i&longs;te&longs;&longs;a, & la corda, <lb/>che è intorno al mezo cerchio <lb/>TVS è eguale alla corda, che è <lb/>intorno al mezo cerchio BCD; <lb/>tolte via dunque le communi PF <lb/>GHT, & SM; &longs;arà la re&longs;tan­<lb/>te MN eguale alle tre BS <lb/>LP DT pre&longs;e in&longs;ieme. </s> <s>ma BS LP DT in&longs;ieme &longs;ono tre volte tanto, quanto<emph.end type="italics"/><lb/> <arrow.to.target n="fig102"></arrow.to.target> <pb/><emph type="italics"/>EQ, & per con&longs;e­<lb/>quenza QR. </s> <s>Lo <lb/>&longs;patio dunque MN <lb/>della traportata po&longs; <lb/>&longs;anza è tre volt<17><lb/>tanto, quanto lo &longs;pa <lb/>tio QR del pe&longs;o <lb/>mo&longs;&longs;o. </s> <s>che era da <lb/>mo&longs;trar&longs;i.<emph.end type="italics"/></s> </p> <figure id="fig102"></figure> <p type="main"> <s><emph type="italics"/>Il tempo ancora di que <lb/>sto mouimento è <lb/>mani&longs;e&longs;to, percio­<lb/>che la po&longs;&longs;anza i&longs;te&longs; <lb/>&longs;a in tempo eguale <lb/>mouerà l'i&longs;te&longs;&longs;o pe­<lb/>&longs;o in i&longs;patio tre co­<lb/>tanto maggiore &longs;en­<lb/>za tali taglie, di <lb/>quel che &longs;arebbe <lb/>con e&longs;&longs;e taglie à que <lb/>&longs;to modo commoda <lb/>te. </s> <s>Lo &longs;patio del <lb/>pe&longs;o mo&longs;&longs;o &longs;enza le <lb/>taglie è eguale allo <lb/>&longs;patio della po&longs;&longs;an­<lb/>za. </s> <s>& in que&longs;to <lb/>modo ritrouaremo <lb/>in tutte il tempo.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="78"/> <p type="head"> <s>PROPOSITIONE XIIII.</s> </p> <p type="main"> <s>Legando la corda d'intorno à tre girelle di due taglie, l'vna del­<lb/>lequali &longs;ia di &longs;opra, & habbia vna &longs;ola girella, & l'altra di &longs;ot­<lb/>to, & ne habbia due, & &longs;ia lega <lb/>ta al pe&longs;o; laqual corda &longs;ia le­<lb/>gata con l'vno de' capi &longs;uoi in <lb/>qualche loco, & l'altro tenu­<lb/>to dalla po&longs;&longs;anza, che moue il <lb/>pe&longs;o: &longs;arà lo &longs;patio cor&longs;o dal­<lb/>la po&longs;&longs;anza, che tira, quattro <lb/>volte tanto, quanto è lo &longs;patio <lb/>del pe&longs;o mo&longs;&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A, &longs;iano le due girelle, i cui cen <lb/>tri<emph.end type="italics"/> K <emph type="italics"/>I della taglia legata al pe&longs;o con <lb/>K<foreign lang="greek">a</foreign>; dimodo, che il pe&longs;o &longs;empre &longs;egua il <lb/>mouimento della taglia in &longs;u&longs;o, ouero in <lb/>giu&longs;o: &longs;ia dapoi la girella il cui centro L <lb/>della taglia appe&longs;a di &longs;opra in <foreign lang="greek">d</foreign>; & &longs;ia la <lb/>corda BCDEFGHZMNO riuolta <lb/>intorno à tutte le girelle, & legata in B; <lb/>& &longs;ia in O la po&longs;&longs;anza, che moue il pe­<lb/>&longs;o A. </s> <s>Dico lo &longs;patio, ilquale la po&longs;&longs;an <lb/>za di O mouendo trapa&longs;&longs;a, e&longs;&longs;ere quat­<lb/>tro volte tanto, quanto lo &longs;patio del pe­<lb/>&longs;o A mo&longs;&longs;o. </s> <s>Mouan&longs;i le girelle della <lb/>taglia legata al pe&longs;o; & mentre il centro <lb/>K è in R, il centro I &longs;ia in S, & il <lb/>pe&longs;o A, cioè il punto <foreign lang="greek">a</foreign> in <foreign lang="greek">b</foreign>: &longs;aranno <lb/>IS KR <foreign lang="greek">ab</foreign> tra &longs;e eguali, & parimen­<lb/>te KI ad e&longs;&longs;a RS eguale: percioche le <lb/>girelle mantengono fra &longs;e la di&longs;tanza me <lb/>de&longs;ima &longs;empre; & K<foreign lang="greek">a</foreign> &longs;arà eguale ad e&longs; <lb/>&longs;a R<foreign lang="greek">b. </foreign></s> <s>&longs;iano condotte per li centri delle <lb/>girelle le linee FHQTECVXNZ <lb/>egualmente distanti dall orizonte, lequa <lb/>li tocchino le corde ne i puntl FH QT<emph.end type="italics"/><lb/> <arrow.to.target n="fig103"></arrow.to.target> <pb/><emph type="italics"/>EC VX NZ che parimente &longs;aranno <lb/>fra loro egualmente di&longs;tanti: & EQ CT <lb/>VN XZ non &longs;olamente fra &longs;e, ma <lb/>ancora ad e&longs;&longs;e IS KR <foreign lang="greek">ab</foreign> &longs;aranno e­<lb/>guali: & mentre li centri KI &longs;ono in <lb/>RS, la po&longs;&longs;anza di O &longs;ia mo&longs;&longs;a in P. </s> <s><lb/>Et percioche la corda BCDEFGHZ <lb/>MNO è eguale alla corda BT<36>QF <lb/>GHX<foreign lang="greek">*u</foreign>VP e&longs;&longs;endo vna corda mede­<lb/>&longs;ima, & le corde d'intorno à mezi cerchi <lb/>T<36>Q X<foreign lang="greek">*u</foreign>V &longs;ono eguali alle corde, che <lb/>&longs;ono d'intorno à CDE ZMN; tolte <lb/>via dunque le communi BT, QFGHX, <lb/>& VO; &longs;arà OP eguale ad e&longs;&longs;e VN <lb/>XZ CT QE pre&longs;e tutte in&longs;ieme. </s> <s>male <lb/>quattro VN ZX CT QE &longs;ono tra&longs;e <lb/>eguali, & in&longs;ieme quattro volte tanto <lb/>quanto KR & <foreign lang="greek">ab. </foreign></s> <s>Per laqual co&longs;a OP <lb/>&longs;arà quattro volte tanto quanto è e&longs;&longs;a <lb/><foreign lang="greek">ab. </foreign></s> <s>Adunque lo &longs;patio della po&longs;&longs;anza <lb/>è quattro volte tanto quanto è lo &longs;palio <lb/>del pe&longs;o. </s> <s>che era da mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig103"></figure> <p type="main"> <s><emph type="italics"/>Et &longs;e la corda in P &longs;arà dauantaggio ri­<lb/>uolta d'intorno ad vn'altra girella ver&longs;o il <lb/><foreign lang="greek">d</foreign>, & la po&longs;&longs;anza mouendo&longs;i in giù mo <lb/>ua in sù il pe&longs;o: &longs;imilmente &longs;i mo&longs;trerà <lb/>lo &longs;patio della po&longs;&longs;anza e&longs;&longs;ere quattro <lb/>volte tanto quanto lo &longs;patio del pe&longs;o.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma&longs;e la corda in B &longs;i riuolgerà d'intorno ad <lb/>vn'altra girella, laqual corda &longs;ilegbi da<emph.end type="italics"/><lb/> <arrow.to.target n="note247"></arrow.to.target> <emph type="italics"/>poi alla taglia di &longs;otto; &longs;arà la po&longs;&longs;anza <lb/>di O, che &longs;o&longs;tiene il pe&longs;o A vn quinto <lb/>dal pe&longs;o. </s> <s>& &longs;e in O &longs;arà la po&longs;&longs;anza, <lb/>che moua il pe&longs;o A; &longs;imilmète &longs;i dimo&longs;tre <lb/>rà lo &longs;patio della po&longs;&longs;anza po&longs;ta in O e&longs;­<lb/>&longs;ere cinque volte tanto quanto lo &longs;patio del pe&longs;o A.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note247"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et &longs;e la corda &longs;i adatterà in modo d'intorno alle girelle, che la po&longs;&longs;anza di O &longs;o&longs;tenen <lb/>te il pe&longs;o &longs;ia vn &longs;e&longs;to del pe&longs;o; & in loco della po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o, &longs;i met­<lb/>ta in O la po&longs;&longs;anza, che lo moua; nell'i&longs;te&longs;&longs;o modo &longs;i mo&longs;trerà lo &longs;patio della po&longs;­<lb/>&longs;anza e&longs;&longs;ere &longs;ei volte tanto quanto lo &longs;patio del pe&longs;o mo&longs;&longs;o. </s> <s>& co&longs;i procedendo in<emph.end type="italics"/> <pb pagenum="79"/><emph type="italics"/>infinito &longs;i troueranno le proportioni dello &longs;patio della po&longs;&longs;anza allo &longs;patio del pe­<lb/>&longs;o mo&longs;&longs;o quanto &longs;i vogliano moltiplici.<emph.end type="italics"/></s> </p> <p type="main"> <s>Et co&longs;i procedendo in infinito &longs;i troueranno le proportioni dello &longs;patio della po&longs;­<lb/>&longs;anza allo &longs;patio del pe&longs;o mo&longs;&longs;o quanto &longs;i vorrà moltiplici. </s> <s>Già è detto che mol <lb/>tiplice è il primo genere delle proportioni nelle quantità paragonate dal mag­<lb/>giore al minore, però qui vuol dire, che con tale regola &longs;i ritroueranno le pro­<lb/>portioni dello &longs;patio del pe&longs;o allo &longs;patio della po&longs;&longs;anza in infinito, <expan abbr="dou&etilde;do">douendo</expan> e&longs;&longs;ere <lb/>lo &longs;patio della po&longs;&longs;anza mouente moltiplice, cioè molte volte maggiore dello <lb/>&longs;patio del pe&longs;o mo&longs;&longs;o, come appare nel pre&longs;ente e&longs;&longs;empio, che è &longs;ei volte più, <lb/>come &longs;ei ad vno; & que&longs;to è il &longs;ignifi cato di moltiplice. </s> </p> <p type="head"> <s>COROLLARIO I.</s> </p> <p type="main"> <s>Da que&longs;te co&longs;e è manife&longs;to, co&longs;i hauer&longs;i il pe&longs;o ver&longs;o la po&longs;&longs;an­<lb/>za, che lo &longs;o&longs;tiene, come lo &longs;patio della po&longs;&longs;anza che moue al­<lb/>lo &longs;patio del pe&longs;o mo&longs;&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Come &longs;e il pe&longs;o A &longs;arà cinque volte tanto quanto la po&longs;&longs;anza di O, che &longs;o&longs;tiene <lb/>il detto pe&longs;o A; &longs;arà anche lo &longs;patio OP della po&longs;&longs;anza mouente il pe&longs;o cin­<lb/>que volte tanto quanto lo &longs;patio <foreign lang="greek">a b</foreign> del pe&longs;o mo&longs;&longs;o.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO II.</s> </p> <p type="main"> <s>E manife&longs;to ancora per le co&longs;e dette, che le girelle della taglia, <lb/>laquale è legata al pe&longs;o, fanno sì, che minore &longs;patio è quello, <lb/>ilquale è de&longs;critto dal pe&longs;o mo&longs;&longs;o, che dalla po&longs;&longs;anza che tira; <lb/>& che in tempo maggiore &longs;i de&longs;criua vn dato &longs;patio eguale, <lb/>che &longs;enza loro: ilche veramente non fanno le girelle della ta­<lb/>glia di &longs;opra. </s> </p> <p type="main"> <s><emph type="italics"/>Mo&longs;trata la proportione moltiplice, che ha il pe&longs;o ver&longs;o la po&longs;&longs;anza, hora &longs;i mo&longs;triper <lb/>lo contrario la proportione moltiplice, che haue la po&longs;&longs;anza ver&longs;o il pe&longs;o.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE XV.</s> </p> <p type="main"> <s>Se la corda &longs;arà inuolta d'intorno alla girella della taglia tenu­<lb/>ta di &longs;opra dalla po&longs;&longs;anza; l'vn capo dellaquale &longs;ia legato in <lb/>qualche loco, ma all'altro &longs;ia appiccato il pe&longs;o, &longs;arà la po&longs;&longs;an­<lb/>za due volte tanto quanto il pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la taglia, che habbia la girella co'l &longs;uo centro A; & &longs;ia il pe&longs;o B legato alla <lb/>corda CDEFG, laquale &longs;ia in uol­<lb/>ta d'intorno alla girella, & alla fine <lb/>legata in G; & &longs;ia la po&longs;&longs;anza, che <lb/>&longs;o&longs;tiene il pe&longs;o in H. </s> <s>Dico, che la <lb/>po&longs;&longs;anza di H è due volte tanto quan <lb/>to il pe&longs;o B. </s> <s>Sia condotta la linea <lb/>DF per lo centro A egualmente di <lb/>&longs;tante dall'orizonte. </s> <s>Percioche dun­<lb/>que la po&longs;&longs;anza di H &longs;o&longs;tiene la ta­<lb/>glia, laquale &longs;o&longs;tiene la girella nel &longs;uo <lb/>centro A, laqual girella &longs;o&longs;liene il pe <lb/>&longs;o; &longs;arà la po&longs;&longs;anza, che &longs;o&longs;tiene la gi­<lb/>rella, come &longs;e fo&longs;&longs;e po&longs;ta in A; &longs;tan­<lb/>do dunque e&longs;&longs;a in A, & il pe&longs;o ap­<lb/>piccato in D, & legato alla corda <lb/>CD; &longs;arà la DF come leua, il cui <lb/>&longs;o&longs;tegno &longs;arà F, il pe&longs;o in D & la<emph.end type="italics"/><lb/> <arrow.to.target n="note248"></arrow.to.target> <emph type="italics"/>po&longs;&longs;anza in A. </s> <s>Ma la po&longs;&longs;anza ver­<lb/>&longs;o il pe&longs;o è come DF ad FA, & <lb/>DF è il doppio di FA: adunque la <lb/>po&longs;&longs;anza di A ouero di H, che è <lb/>l'i&longs;te&longs;&longs;o, &longs;arà due volte tanto, quanto il <lb/>pe&longs;o B. che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note248"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo nella leua.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Oltre à ciò occorre à con&longs;iderare, &longs;tando ferme tutte que&longs;te co&longs;e, che egli è l'i&longs;te&longs;&longs;o, e&longs;­<lb/>&longs;endo vna corda &longs;ola CDEFG in que&longs;to modo inuolta d'intorno alla girella, co­<lb/>me &longs;e fo&longs;&longs;ero due corde CDFG legate nella leua, ouero nella bilancia DF.<emph.end type="italics"/></s> </p> <p type="head"> <s>Altramente.</s> </p> <p type="main"> <s><emph type="italics"/>Stando le mede&longs;ime co&longs;e, &longs;e in G &longs;o&longs;&longs;e appiccato un pe&longs;o K eguale al pe&longs;o B, li pe&longs;i <lb/>BK pe&longs;erebbono egualmente nella bilancia DF, il cui centro A. </s> <s>Ma la po&longs;­<lb/>&longs;anza di H, laquale &longs;o&longs;tiene i pe&longs;i BK è eguale ad ambidue pre&longs;i in&longs;ieme, & i <lb/>pe&longs;i BK &longs;ono due volte tanto quanto è e&longs;&longs;o B. </s> <s>Adunque la po&longs;&longs;anza di H &longs;a­<lb/>rà due volte tanto quanto è il B. & percioche la corda legata in G non fa al­<lb/>tro niente, &longs;e non che &longs;o&longs;tiene il pe&longs;o B, che non di&longs;cenda, laqual co&longs;a parimente <lb/>fà il pe&longs;o K appiccato in G: la po&longs;&longs;anza dunque di H, che &longs;o&longs;tiene il pe&longs;o B, <lb/>e&longs;&longs;endo la corda legata in G, è due volte tanto quanto il pe&longs;o B. che bi&longs;ognaua <lb/>mostrare.<emph.end type="italics"/></s> </p> <pb pagenum="80"/> <p type="head"> <s>PROPOSITIONE XVI.</s> </p> <p type="main"> <s>Po&longs;te le co&longs;e i&longs;te&longs;&longs;e, &longs;e in H &longs;arà la po&longs;&longs;anza che moue il pe&longs;o, <lb/>mouerà ella con la leua egualmente di&longs;tante dall'orizonte. </s> </p> <p type="main"> <s><emph type="italics"/>Que&longs;to etiandio &longs;i mo&longs;trerà, co­<lb/>me è detto di &longs;opra. </s> <s>Moua&longs;i <lb/>la girella in sù, & habbia il <lb/>&longs;ito di MNO, il cui centro <lb/>L: & per L &longs;ia condotta la <lb/>linea MLO egualmente di­<lb/>&longs;tante da e&longs;&longs;a DF, & dall'o <lb/>rizonte. </s> <s>& percioche le cor­<lb/>de toccano il cerchio MON <lb/>ne i punti MO; però e&longs;&longs;en­<lb/>do che la po&longs;&longs;anza di A, oue <lb/>ro di H, che è l'i&longs;te&longs;&longs;o, mo­<lb/>ua il pe&longs;o B appiccato in D <lb/>con la leua DF, il cui &longs;o&longs;te­<lb/>gno è F; &longs;empre rimarrà da <lb/>uantaggio vn'altra leua, co­<lb/>me MO egualmente di&longs;tan­<lb/>te dall'orizonte, di modo che <lb/>&longs;empre la po&longs;&longs;anza moua il pe <lb/>&longs;o, &longs;tando la leua egualmente <lb/>di&longs;tante dall'orizonte, il cui <lb/>&longs;o&longs;tegno &longs;empre è nella linea <lb/>OG, & il pe&longs;o in MC, & <lb/>la po&longs;&longs;anza nel centro della <lb/>girella.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s>Po&longs;te le co&longs;e mede&longs;ime, lo &longs;patio del pe&longs;o mo&longs;&longs;o è due volte tan <lb/>to quanto lo &longs;patio della po&longs;&longs;anza, che moue. </s> </p> <p type="main"> <s><emph type="italics"/>Sia mo&longs;&longs;a la girella dal centro A fin al centro L; & il pe&longs;o B, cioè il punto e, <lb/>nell'i&longs;te&longs;&longs;o tempo &longs;ia mo&longs;&longs;o nel P; & la po&longs;&longs;anza di H fin in K; farà AH <lb/>ad e&longs;&longs;a LK eguale, & <lb/>AL ad e&longs;&longs;a HK: & <lb/>percioche le corda CDE <lb/>FG eguale è alla corda <lb/>PMNOG, peroche è <lb/>vna corda i&longs;te&longs;&longs;a, & la <lb/>corda d'intorno al mezo <lb/>cerchio MNO eguale è <lb/>alla corda d'intorno al me <lb/>zo cerchio DEF: tolte <lb/>via dunque le communi <lb/>corde DP FG, &longs;arà <lb/>PC eguale à DM FO <lb/>pre&longs;e in&longs;ieme, lequali cor­<lb/>de &longs;ono due volte tanto <lb/>quanto è e&longs;&longs;a AL & per <lb/>con&longs;eguenza e&longs;&longs;a HK. </s> <s><lb/>Lo &longs;patio dunque del pe­<lb/>&longs;o mo&longs;&longs;o CP è due vol­<lb/>te tanto, quanto è lo &longs;pa­<lb/>tio della po&longs;&longs;anza HK. <lb/>che bi&longs;ognaua mostrare.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>COROLLARIO</s> </p> <p type="main"> <s>Da que&longs;to è manife&longs;to, l'i&longs;te&longs;&longs;o pe&longs;o e&longs;&longs;ere tirato dalla i&longs;te&longs;&longs;a po&longs; <lb/>&longs;anza in tempo eguale per due volte tanto &longs;patio con la taglia <lb/>in que&longs;to modo accommodata, che &longs;enza taglia; pur che i <lb/>mouimenti di e&longs;&longs;a po&longs;&longs;anza &longs;iano eguali in velocità. </s> </p> <p type="main"> <s><emph type="italics"/>Percioche lo &longs;patio del pe&longs;o mo&longs;&longs;o &longs;enza taglia è vguale allo &longs;patio della po&longs;&longs;anza.<emph.end type="italics"/></s> </p> <pb pagenum="81"/> <p type="main"> <s><emph type="italics"/>Che &longs;e la corda &longs;arà in G riuolta d'intorno ad vn'altra girella, il cia centro K; & <lb/>&longs;ia la taglia di cotale girella attaccata di &longs;otto, laquale non habbia alcuno altro <lb/>mouimento, &longs;e non il libero riuolgimento della girella d'intorno all'a&longs;&longs;etto &longs;uo; & <lb/>la corda &longs;i leghi in M; &longs;arà <lb/>la po&longs;&longs;anza di H che &longs;o&longs;tiene <lb/>il pe&longs;o B. &longs;imilmente due vol <lb/>te tanto, quanto è e&longs;&longs;o pe&longs;o. </s> <s>il <lb/>che per certo è manife&longs;to, con­<lb/>cio&longs;ia, che egli &longs;ia in tutto vna <lb/>co&longs;a i&longs;te&longs;&longs;a, &longs;e ouero la corda &longs;ia <lb/>in M ouero in G legata, per­<lb/>cioche la girella del centro K <lb/>non fà nulla, & è totalmente <lb/>inutile.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma &longs;e la po&longs;&longs;anza che &longs;o&longs;tiene il <lb/>pe&longs;o B &longs;arà in M, & la ta <lb/>glia di &longs;opra &longs;ia appiccata in <lb/>sù; &longs;arà la po&longs;&longs;anza di M e­<lb/>guale al pe&longs;o B.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Percioche la po&longs;&longs;anza di G, che<emph.end type="italics"/> <arrow.to.target n="note249"></arrow.to.target><lb/><emph type="italics"/>&longs;o&longs;tiene il pe&longs;o B è eguale al <lb/>pe&longs;o B; & ad e&longs;&longs;a po&longs;&longs;anza <lb/>di G è eguale la po&longs;&longs;anza di <lb/>L; percioche GL è leua, il <lb/>cui &longs;o&longs;iegno è K; & la di­<lb/>&longs;tanza GK è eguale alla di&longs;tan <lb/>za KL; &longs;arà dunque la po&longs;­<lb/>&longs;anza di L, ouero (che è il me <lb/>de&longs;imo,) di M eguale al pe&longs;o B.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note249"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Que&longs;to tale mouimento &longs;i fà nel­<lb/>le leue DF LG i cui &longs;o&longs;tegni <lb/>&longs;ono KA, & il pe&longs;o in D, & la po&longs;&longs;anza in F; ma nella leua LG la po&longs;&longs;an <lb/>za &longs;tà in L, & il pe&longs;o come &longs;e fu&longs;&longs;e in G.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Se poi &longs;arà in M la po&longs;&longs;anza, che moue il pe&longs;o, & &longs;i tra&longs;porti, la po&longs;&longs;anza in N <lb/>& il pe&longs;o &longs;ia mo&longs;&longs;o fin ad O; &longs;arà lo &longs;patio MN della po&longs;&longs;anza eguale allo &longs;patio <lb/>di CO pe&longs;o; percioche e&longs;&longs;endo la corda MLGFDC eguale alla corda NLG <lb/>FDO, peroche è vna i&longs;te&longs;&longs;a corda; leuata via la commune MLGFDO, &longs;arà lo <lb/>&longs;patio MN della po&longs;&longs;anza eguale allo &longs;patio CO del pe&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et &longs;e la corda in M &longs;arà inuolta intorno à più girelle, &longs;empre la po&longs;&longs;anza, che in vno <lb/>delli &longs;uoi e&longs;tremi &longs;o&longs;terrà il pe&longs;o &longs;arà eguale ad e&longs;&longs;o pe&longs;o: & gli &longs;patij del pe&longs;o, & <lb/>della po&longs;&longs;anza che moue &longs;empre &longs;i mo&longs;treranno e&longs;&longs;ere eguali.<emph.end type="italics"/></s> </p> <pb/> <p type="head"> <s>PROPOSITIONE XVII.</s> </p> <p type="main"> <s>Se à cia&longs;cuna delle due girelle di due taglie, l'vna delle quali &longs;ia <lb/>&longs;o &longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra &longs;ia po&longs;ta di &longs;otto, & <lb/>iui attaccata, &longs;i condurrà intorno la corda; con l'vno de' &longs;uoi <lb/>capi legato alla taglia di &longs;opra, & l'altro appiccato al pe&longs;o; la <lb/>po&longs;&longs;anza &longs;arà tre volte tanto quanto il pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la girella co'l centro A della <lb/>taglia all accaia di &longs;otto; & &longs;ia la <lb/>corda BCDEFG inuolta intor <lb/>no non &longs;olamente à cote&longs;ta girel <lb/>la, ma etiandio alla girella della <lb/>taglia di &longs;opra, che ha il centro K; <lb/>& &longs;ia la cordalegata in B della <lb/>taglia di &longs;opra; & in G &longs;ia at­<lb/>taccato il pe&longs;o H; & la po&longs;&longs;an­<lb/>zain L &longs;o&longs;tenga il pe&longs;o H. </s> <s>Di­<lb/>co che la po&longs;&longs;anza in L ètre vol <lb/>te tanto quanto il pe&longs;o H, per­<lb/>cioche &longs;e fo&longs;&longs;ero due po&longs;&longs;anze, che <lb/>&longs;o&longs;tenne&longs;&longs;ero il pe&longs;o H vna in K, <lb/>& l'altra in B, &longs;arebbono ambe­<lb/>due in&longs;ieme tre volte tanto quan-<emph.end type="italics"/><lb/> <arrow.to.target n="note250"></arrow.to.target> <emph type="italics"/>to il pe&longs;o H: percioche la po&longs;&longs;an <lb/>za in K è due volte tanto quan­<lb/>to il pe&longs;o H, & la po&longs;&longs;anza in <lb/>B è eguale ad e&longs;&longs;o pe&longs;o. </s> <s>& per <lb/>cioche la &longs;ola po&longs;&longs;anza in L è <lb/>eguale ad ambedue le po&longs;&longs;anze in <lb/>KB, peroche la po&longs;&longs;anza in L &longs;o­<lb/>&longs;tiene sì la po&longs;&longs;anza po&longs;ta in K, <lb/>come la po&longs;&longs;anza po&longs;ta in B; & <lb/>la detta po&longs;&longs;anza in L fa l'i&longs;te&longs;&longs;o, <lb/>come &longs;e fu&longs;&longs;ero due po&longs;&longs;anze, l'v­<lb/>na in K & l'altra in B. </s> <s>Sarà <lb/>dunque tre volte tanto la po&longs;&longs;an­<lb/>za in L quanto il pe&longs;o H. </s> <s>Che <lb/>bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note250"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo. </s> <s>Nella prece dente.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="82"/> <p type="main"> <s>Ma &longs;e in L &longs;arà la po&longs;&longs;anza, che moue il pe&longs;o. </s> <s>Dico lo &longs;patio <lb/>del pe&longs;o mo&longs;&longs;o e&longs;&longs;ere tre volte tanto, quanto lo &longs;patio della <lb/>po&longs;&longs;anza mo&longs;&longs;a. </s> </p> <p type="main"> <s><emph type="italics"/>Moua&longs;i il centro della girella <lb/>K fin ad M, lo &longs;patio <lb/>delquale mouimento è ve­<lb/>ramente eguale allo &longs;patio <lb/>della po&longs;&longs;anza mo&longs;&longs;a, co­<lb/>me è detto di &longs;opra: &<emph.end type="italics"/> <arrow.to.target n="note251"></arrow.to.target><lb/><emph type="italics"/>quando K &longs;arà in M, B <lb/>&longs;arà in N, & NB &longs;a <lb/>rà eguale ad MK; & <lb/>mentre K è in M, fia il <lb/>pe&longs;o H, cioè il punto G <lb/>mo&longs;&longs;o in O; & per MK <lb/>&longs;iano condotte le linee EF <lb/>PQ egualmente di&longs;tanti <lb/>dall'orizonte; &longs;arà cia&longs;cu­<lb/>na delle EP BN FQ <lb/>eguale ad e&longs;&longs;a KM. </s> <s>Et <lb/>perciochela co da BCD <lb/>EFG eguale è alla corda <lb/>NCDPQO; e&longs;&longs;endo <lb/>vna mede&longs;ima corda; & <lb/>la corda po&longs;ta intorno al <lb/>mezo cerchio ERF e­<lb/>guale è alla corda po&longs;ta in <lb/>torno al mezo cerchio <lb/><expan abbr="PSq;">PSque</expan> tolte via dunque <lb/>le corde communi BC <lb/>DE, & FO, &longs;arà OG <lb/>eguale alle tre corde QF <lb/>NB PE pre&longs;e in&longs;ieme. </s> <s><lb/>ma QF NB PE in­<lb/>&longs;ieme &longs;ono tre volte tanto <lb/>quanto MK, cioè lo &longs;pa­<lb/>tio della po&longs;&longs;anza mo&longs;&longs;a; <lb/>lo &longs;patio dunque GO del <lb/>pe&longs;o H mo&longs;&longs;o, è tre vol­<lb/>te tanto quanto è lo &longs;pa­<lb/>tio della po&longs;&longs;anza mo&longs;&longs;a. </s> <s><lb/>che bi&longs;egn<gap/>a mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note251"></margin.target><emph type="italics"/>Nella precede<gap/>e.<emph.end type="italics"/></s> </p> <figure></figure> <pb/> <p type="head"> <s>PROPOSITIONE XVIII.</s> </p> <p type="main"> <s>Se ad ambedue le girelle delle due taglie: l'vna delle quali &longs;ia &longs;o <lb/>&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra &longs;ia po&longs;ta di &longs;otto, & <lb/>iui attaccata, &longs;arà inuolta intorno la corda; con l'vno de' ca­<lb/>pi &longs;uoi in qualche luogo legato, ma non già nella taglia di &longs;o­<lb/>pra, & all'altro &longs;ia appiccato il pe&longs;o; la po&longs;&longs;anza &longs;arà quattro <lb/>volte tanto quanto il pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la taglia di &longs;otto, che habbia due gi­<lb/>relle con li centri &longs;uoi AB; & &longs;ia <lb/>la taglia di &longs;opra, che &longs;imilmente hab­<lb/>bia due girelle con li centri &longs;uoi CD: <lb/>& &longs;ia la corda EFGHKLMNOP <lb/>riuolta d'intorno à tutte le girelle, che <lb/>&longs;ia legata poi in E, & &longs;ia appicca­<lb/>to in P il pe&longs;o Q: & &longs;ia la po&longs;&longs;an­<lb/>za in R. </s> <s>Dico la po&longs;&longs;anza di R e&longs;­<lb/>&longs;ere quattro volte tanto quanto il pe­<lb/>&longs;o Q: concio&longs;ia che &longs;e &longs;i intenderan-<emph.end type="italics"/><lb/> <arrow.to.target n="note252"></arrow.to.target> <emph type="italics"/>no due po&longs;&longs;anze, l'vnain K & l'al­<lb/>train D, la po&longs;&longs;anza in K che &longs;o <lb/>&longs;tiene il pe&longs;o Q con la corda KLM <lb/>NOP &longs;arà eguale al pe&longs;o; & &longs;aran <lb/>no le due po&longs;&longs;anze in&longs;ieme l'vna in D <lb/>& l'altra in K &longs;ostenenti il pe&longs;o Q <lb/>tre volte tanto quanto l'i&longs;te&longs;&longs;o pe&longs;o. </s> <s><lb/>Ma la po&longs;&longs;anza di C è due volte tan <lb/>to quanto la po&longs;&longs;anza di K, & per con<emph.end type="italics"/><lb/> <arrow.to.target n="note253"></arrow.to.target> <emph type="italics"/>&longs;equenza del pe&longs;o <expan abbr="q;">que</expan> peroche egli è <lb/>la me de&longs;ima co&longs;a, come &longs;e in K fo&longs;&longs;e <lb/>appiccato vn pe&longs;o eguale al pe&longs;o Q, <lb/>delquale è due volte tanto la po&longs;&longs;anza <lb/>di C. </s> <s>Adunque due po&longs;&longs;anze po&longs;te <lb/>in DC &longs;ono quattro volte tanto quan <lb/>to è il pe&longs;o <expan abbr="q.">que</expan> & concio&longs;ia, che la <lb/>po&longs;&longs;anza di R &longs;o&longs;tenga con le girelle <lb/>il pe&longs;o Q, &longs;arà la po&longs;&longs;anza di R co­<lb/>me &longs;e fo&longs;&longs;ero due po&longs;&longs;anze l'vna in D<emph.end type="italics"/><lb/> <arrow.to.target n="fig104"></arrow.to.target> <pb pagenum="83"/><emph type="italics"/>& l'altra in C: & l'vna, & l'altra in&longs;ieme &longs;o&longs;tene&longs;&longs;e il pe&longs;o <expan abbr="q.">que</expan> La po&longs;&longs;anza <lb/>dunque di R è quattro volte tanto quanto il pe&longs;o <expan abbr="q.">que</expan> che bi&longs;ognaua dimo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note252"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note253"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure id="fig104"></figure> <p type="head"> <s>COROLLARIO</s> </p> <p type="main"> <s>Dalla qual co&longs;a è manife&longs;to, che &longs;e la corda &longs;arà legata in G, & <lb/>riuolta d'intorno alle girelle, i cui centri &longs;ono BCD; &longs;arà <lb/>la po&longs;&longs;anza di R che &longs;o&longs;tiene quat <lb/>tro volte tanto, &longs;imilmente quan­<lb/>to il pe&longs;o <expan abbr="q.">que</expan> Percioche la girel­<lb/>la il cui centro è A non fà nulla. </s> </p> <p type="main"> <s>Che &longs;e la po&longs;&longs;anza mouènte il pe&longs;o &longs;a <lb/>rà in R. </s> <s>Dico lo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o e&longs;&longs;ere quattro volte tanto <lb/>quanto lo &longs;patio della po&longs;&longs;anza. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Siano mo&longs;&longs;i i centri CD delle girelle fin ad ST; <lb/>&longs;aranno per le co&longs;e di &longs;opra dette CS DT <lb/>eguali allo &longs;patio della po&longs;&longs;anza; & per SDT <lb/>&longs;iano condotte le linee HK VX NO <foreign lang="greek">*u</foreign>Z <lb/>egualmente di&longs;tanti dall'orizonte; & mentre <lb/>li centri CD &longs;ono in ST, &longs;ia il pe&longs;o Q, <lb/>cioè il punto P mo&longs;&longs;o in <36>. </s> <s>& percioche <lb/>la corda EFGHKLMNOP eguale è al <lb/>la corda EFGVXLM<foreign lang="greek">*u</foreign>Z<36>; e&longs;&longs;endo vna <lb/>mede&longs;ima corda: & le corde po&longs;te d'intorno à <lb/>mezi cerchi NIOH<foreign lang="greek">a</foreign>K &longs;iano egua i alle cor <lb/>de, lequali &longs;ono intorno à i mezi cerchi <foreign lang="greek">*ud</foreign>Z <lb/>V<foreign lang="greek">b</foreign>X; tolte via dunque le communi EFGH <lb/>KLMN & O<36>; &longs;arà P<36> eguale ad e&longs;&longs;e <lb/>N<foreign lang="greek">*u</foreign> ZO VH XK in&longs;ieme pre&longs;e, ma le quat <lb/>tro N<foreign lang="greek">*u</foreign> ZO VH XK tutte in&longs;ieme &longs;ono <lb/>quattro volte tanto quanto DT cioè lo &longs;pa­<lb/>tio della po&longs;&longs;anza. </s> <s>Lo &longs;patio dunque PQ del <lb/>pe&longs;o è quattro volte tanto quanto lo &longs;patio <lb/>della po&longs;&longs;anza. </s> <s>che era da mo&longs;trar&longs;i.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma &longs;e la corda &longs;a <lb/>rà rilegata in <lb/>E della taglia <lb/>di &longs;opra, & la <lb/>po&longs;&longs;anza di R <lb/>&longs;o&longs;tenga il pe­<lb/>&longs;o <expan abbr="q;">que</expan> &longs;arà la <lb/>po&longs;&longs;anza di R <lb/>cinque volte <lb/>tanto quanto <lb/>il pe&longs;o <expan abbr="q.">que</expan> & <lb/>&longs;e in R &longs;arà <lb/>la po&longs;&longs;anza, <lb/>che moue il pe <lb/>&longs;o &longs;arà lo &longs;pa <lb/>tio del pe&longs;o <lb/>mo&longs;&longs;o cinque <lb/>volte tanto, <lb/>quanto lo &longs;pa­<lb/>tio della po&longs;­<lb/>&longs;anza. </s> <s>Lequa­<lb/>li co&longs;e tutte &longs;i <lb/>dimo&longs;trer anno <lb/>con modo &longs;imi <lb/>le, come nelle <lb/>precedenti è <lb/>&longs;tato fatto.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="84"/> <p type="main"> <s><emph type="italics"/>Ma &longs;e la po&longs;&longs;anza di R &longs;o­<lb/>&longs;tene&longs;&longs;e il pe&longs;o Q hauen­<lb/>do la taglia tre girelle, i <lb/>cui centri &longs;iano ABC; & <lb/>&longs;ia vn'altra taglia di &longs;otto, <lb/>che habbia due, ò tre girel­<lb/>le, i cui centri &longs;iano DEF; <lb/>& &longs;ia la corda riuolta d'in <lb/>torno à tutte le girelle, & <lb/>&longs;ia legatain G ouero in H; <lb/>&longs;imilmente mo&longs;trera&longs;&longs;i la <lb/>po&longs;&longs;anza di R e&longs;&longs;ere &longs;ei <lb/>volte tanto quanto il pe&longs;o <lb/><expan abbr="q.">que</expan> & &longs;e in R &longs;arà la <lb/>forza mouente il pe&longs;o, &longs;i <lb/>mo&longs;trer<gap/> lo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o e&longs;&longs;ere &longs;ei volte tan­<lb/>to quanto lo &longs;patio della <lb/>po&longs;&longs;anza.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Et &longs;e la corda &longs;arà legata in <lb/>K della taglia di &longs;opra, & <lb/>in R &longs;ia la po&longs;&longs;anza che <lb/>&longs;o&longs;tiene il pe&longs;o; con modo <lb/>&longs;imile &longs;i prouerà la po&longs;&longs;an­<lb/>za di R e&longs;&longs;ere &longs;ette volte <lb/>tanto quanto il pe&longs;o <expan abbr="q.">que</expan><emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et &longs;e in R &longs;arà la po&longs;&longs;anza <lb/>che moue, &longs;i mo&longs;trerà lo &longs;pa <lb/>tio del pe&longs;o Q e&longs;&longs;ere &longs;ette <lb/>volte tanto quanto lo &longs;pa­<lb/>tio della po&longs;&longs;anza. </s> <s>& co&longs;t <lb/>in infinito ogni proportio­<lb/>ne molteplice della po&longs;&longs;an­<lb/>za ver&longs;o il pe&longs;o potra&longs;&longs;i <lb/>trouare. </s> <s>& &longs;i mo&longs;trerà <lb/>&longs;empre, co&longs;i e&longs;&longs;ere il pe&longs;o <lb/>ver&longs;o la po&longs;&longs;anza che lo &longs;o­<lb/>&longs;tiene, come lo &longs;patio della <lb/>po&longs;&longs;anza che moue il pe&longs;o, <lb/>allo &longs;patio del pe&longs;o mo&longs;&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Hor il mouimento <lb/>delle leue delle gi <lb/>relle in que&longs;te &longs;i fà <lb/>in cotal modo, <lb/>cioè le leue delle <lb/>girelle della taglia <lb/>di &longs;opra &longs;i mouo­<lb/>no, come è detto, <lb/>nella decima&longs;e&longs;ta <lb/>di que&longs;to; cioè han <lb/>no il &longs;o&longs;tegno nel­<lb/>le &longs;tremità, la po&longs;­<lb/>&longs;anza nel mezo, <lb/>& il pe&longs;o nell'al­<lb/>tra &longs;tremità ap­<lb/>piccato. </s> <s>Ma le <lb/>leue della taglia di <lb/>&longs;otto hanno il &longs;o­<lb/>&longs;tegno nel mezo, <lb/>& il pe&longs;o, & la <lb/>po&longs;&longs;anzanelle &longs;tro <lb/>mità.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="85"/> <p type="head"> <s>COROLLARIO</s> </p> <p type="main"> <s>In que&longs;te co&longs;e è manife&longs;to, che le girelle della taglia di &longs;opra &longs;o­<lb/>no cagione, che il pe&longs;o &longs;i moua da po&longs;&longs;anza maggio re di e&longs;&longs;o <lb/>pe&longs;o, & per maggiore &longs;patio di quel che è lo &longs;patio di e&longs;&longs;a po&longs; <lb/>&longs;anza, & per eguale in manco tempo: co&longs;a che veramente <lb/>non fanno le girelle della taglia di &longs;otto. </s> </p> <p type="main"> <s><emph type="italics"/>In altro modo ancora po&longs;&longs;iamo ritrouare que&longs;ta proportione moltiplice della po&longs;&longs;an­<lb/>za ver&longs;o il pe&longs;o.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE XIX.</s> </p> <p type="main"> <s>Se à cia&longs;cuna delle girelle dell'vna, & l'altra delle due taglic, l'v­<lb/>na delle quali &longs;ia appiccata di &longs;opra, & l'altra di &longs;otto ritenu­<lb/>ta dalla po&longs;&longs;anza, che &longs;o&longs;tiene, &longs;i riuolga intorno la corda; con <lb/>l'vno de' capi &longs;uoi legato in qualche loco, & con l'altro attac­<lb/>cato al pe&longs;o: la po&longs;&longs;anza &longs;arà due volte tanto quanto il pe&longs;o. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia la girella della taglia appiccata di &longs;opra, il cui centro &longs;ia A; & BCD &longs;ia del­<lb/>la taglia di &longs;otto; &longs;ia dapoi la corda EBCDFGHL rilegata in E; & in L &longs;ia <lb/>appiccato il pe&longs;o M; & &longs;ia la po&longs; <lb/>&longs;anza che &longs;ostiene il pe&longs;o M po&longs;ta in <lb/>N. </s> <s>Dico la po&longs;&longs;anza di N e&longs;&longs;ere<emph.end type="italics"/><lb/> <arrow.to.target n="note254"></arrow.to.target> <emph type="italics"/>due volte tanto quanto il pe&longs;o M. </s> <s>Per <lb/>cioche e&longs;&longs;endo &longs;tato di &longs;opra mo&longs;trato <lb/>la po&longs;&longs;anza di L, laquale per gratia <lb/>di e&longs;&longs;empio, &longs;o&longs;teaga il pe&longs;o O ap­<lb/>piccato in N, e&longs;&longs;ere la metà meno di <lb/>e&longs;&longs;o pe&longs;o; adunque la po&longs;&longs;anza di N, <lb/>che è eguale al pe&longs;o O &longs;o&longs;tenirà il pe­<lb/>&longs;o M, che è eguale alla po&longs;&longs;anza di L; <lb/>& &longs;arà detta po&longs;&longs;anza due volte tan­<lb/>to quanto il pe&longs;o M. che bi&longs;ognaua <lb/>mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note254"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>Altramente.</s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;te le co&longs;e i&longs;te&longs;&longs;e. </s> <s>Percioche la po&longs;&longs;an­<lb/>za di F, ouero di D, che è l'i&longs;te&longs;&longs;o,<emph.end type="italics"/><lb/> <arrow.to.target n="note255"></arrow.to.target> <emph type="italics"/>è eguale al pe&longs;o M: & BD è vna <lb/>leua, il cui &longs;o&longs;tegno è B, & la po&longs;­<lb/>&longs;anza di N è come &longs;e ella fo&longs;&longs;e nel <lb/>mezo della leua, & il pe&longs;o eguale ad <lb/>e&longs;&longs;o M &longs;tà come &longs;e egli fu&longs;&longs;e in D <lb/>per cau&longs;a della corda FD, che è l'i­<lb/>&longs;te&longs;&longs;o, come &longs;e BCD fo&longs;&longs;e la girella <lb/>della taglia di &longs;opra, & il pe&longs;o fo&longs;&longs;e <lb/>appiccato nella corda DF, &longs;i come <lb/>nella decimaquinta, & nella decima­<lb/>&longs;e&longs;ta è detto. </s> <s>La po&longs;&longs;anza dunque di <lb/>N è due volte tanto, quanto il pe&longs;o <lb/>M. che era da mo&longs;trar&longs;i.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note255"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma &longs;e in N &longs;arà la po&longs;&longs;anza, che moue <lb/>il pe&longs;o M, &longs;arà lo &longs;patio del pe&longs;o M <lb/>due volte tanto quanto la po&longs;&longs;anza po&longs;ta in N, ilche è mani&longs;e&longs;to dalla duodecima <lb/>di que&longs;to; percioche lo &longs;patio del punto L che inchina in giu&longs;o, è due volte tanto <lb/>quanto lo &longs;patio di N che và in &longs;u&longs;o; &longs;arà dunque per lo contrario lo &longs;patio del­<lb/>la po&longs;&longs;anza di N che inchina in giù la metà meno dello &longs;patio del pe&longs;o M mo&longs;­<lb/>&longs;a all'in sù.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Hor &longs;i come dalla terza, dalla quinta, & dalla &longs;ettima di que&longs;to &c. </s> <s>&longs;i po&longs;&longs;ono rac-<emph.end type="italics"/> <pb pagenum="86"/><emph type="italics"/>cogliere le ragioni del pe&longs;o O, &longs;iano quanto &longs;i voglia molteplici ad e&longs;&longs;a po&longs;&longs;anza <lb/>po&longs;ta in L, con l'i&longs;te&longs;&longs;o modo parimente &longs;i potranno mo&longs;trare le ragioni quanto <lb/>&longs;i voglia molteplici della po&longs;&longs;anza po&longs;ta in N, che &longs;o&longs;tiene il pe&longs;o M. & co&longs;i <lb/>dalla decnnaterza, & dalla decimaquarta &longs;i mo&longs;treranno le ragioni quanto &longs;i voglia <lb/>molteplici allo &longs;patio del pe&longs;o M, allo &longs;patio della po&longs;&longs;anza po&longs;ta in N.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Si potrà ancora dalla decima&longs;ettima, & dalla decimaottaua di que&longs;to ritrouare la <lb/>proportione molteplice, laquale ha la po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o ver&longs;o l'i&longs;te&longs;&longs;o <lb/>pe&longs;o, &longs;i come la proportione della po&longs;&longs;anza di N al pe&longs;o M &longs;i dimo&longs;traua nel­<lb/>la propo&longs;itione decimaquinta, & decima&longs;e&longs;ta: & &longs;i trouerà co&longs;i e&longs;&longs;ere il pe&longs;o <lb/>alla po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o; come lo &longs;patio della po&longs;&longs;anza, che moue allo <lb/>&longs;patio del pe&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Li mouimenti delle leue in que&longs;te &longs;i fà in cotal modo, cioè le leue delle girelle della ta­<lb/>glia di &longs;otto &longs;i mouono, come della leua BD, laquale &longs;i moue, come &longs;e B fo&longs;&longs;e il <lb/>&longs;o&longs;tegno, & il pe&longs;o &longs;te&longs;&longs;e in D, & la po&longs;&longs;anza nel mezo. </s> <s>Ma le leue delle girel­<lb/>le della taglia di &longs;opra &longs;i mouono, come FH, il cui &longs;o&longs;tegno è nel mezo, il pe&longs;o in <lb/>H & la po&longs;&longs;anza in F.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Da que&longs;to è manife&longs;to, che le girelle della taglia di &longs;otto in que­<lb/>&longs;te fanno effetto tale, che il pe&longs;o vien mo&longs;&longs;o da po&longs;&longs;anza mag­<lb/>giore, di quel che &longs;ia e&longs;&longs;o pe&longs;o, & per maggiore &longs;patio dello <lb/>&longs;patio di e&longs;&longs;a po&longs;&longs;anza, & per eguale in manco tempo. </s> <s>Co&longs;a <lb/>che non fanno già le girelle della taglia di &longs;opra. </s> </p> <p type="main"> <s><emph type="italics"/>Cono&longs;ciute le proportioni molteplici, hor egli è da accostar&longs;i alle &longs;opra particolari.<emph.end type="italics"/></s> </p> <p type="main"> <s>Cono&longs;ciute le proportioni molteplici, già egli è da venire alle &longs;opraparticolari. </s> <s>Il <lb/>genere &longs;opraparticolare è il &longs;econdo propo&longs;to di &longs;opra, quando cio è &longs;i paragona <lb/>vna quantità maggiore ver&longs;o vna minore &longs;i fattamente, che e&longs;&longs;a maggiore con­<lb/>tenga la minore vna ò piu volte, & di piu parte di e&longs;&longs;a, che la pos&longs;i numerare in­<lb/>teramente: come per e&longs;&longs;empio, il tre contiene il due vna volta, & più la metà di <lb/>e&longs;&longs;o due, cioè vno, ilquale puote numerare il tre. </s> <s>Intende dunque l'autore <expan abbr="d'in-ue&longs;tīgare">d'in­<lb/>ue&longs;tingare</expan> la proportione &longs;opraparticolare, che hà il pe&longs;o alla po&longs;&longs;anza. </s> </p> <pb/> <p type="head"> <s>PROPOSITIONE XX.</s> </p> <p type="main"> <s>Se à cia&longs;cuna delle girelle dell'vna & l'altra delle due taglie, l'v­<lb/>na delle quali &longs;ia &longs;o&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & di &longs;otto <lb/>&longs;ia po&longs;ta, & legata al pe&longs;o, &longs;arà inuolta d'intorno la corda; <lb/>con l'vno de' &longs;uoi capi legato in qualche loco, & l'altro attac­<lb/>cato alla taglia di &longs;otto; il pe&longs;o &longs;arà vna volta & meza tanto <lb/>quanto la po&longs;&longs;anza. </s> </p> <p type="main"> <s><emph type="italics"/>Sia ABC la girella della taglia di &longs;opra, & DEF <lb/>quella della taglia di &longs;otto legata al pe&longs;o G; & <lb/>&longs;ia la corda HABCDEFK inuolta d'intorno <lb/>alle, girellel aqual corda &longs;ia legata in K, & in H <lb/>alla taglia di &longs;otto; & &longs;ia in L la po&longs;&longs;anza che <lb/>&longs;o&longs;tiene il pe&longs;o G. </s> <s>Dico, che il pe&longs;o è vna volta <lb/>& meza tanto quanto la po&longs;&longs;anza. </s> <s>Hor percio-<emph.end type="italics"/><lb/> <arrow.to.target n="note256"></arrow.to.target> <emph type="italics"/>che l'vna, & l'altra corda CD AH &longs;o&longs;tiene la <lb/>terza parte del pe&longs;o G; &longs;arà ogn'vna delle po&longs;­<lb/>&longs;anze po&longs;te in DH vn terzo del pe&longs;o G; alle <lb/>quali tutte pre&longs;e in&longs;ieme è eguale la po&longs;&longs;anza di<emph.end type="italics"/><lb/> <arrow.to.target n="note257"></arrow.to.target> <emph type="italics"/>L: peroche la detta po&longs;&longs;anza di L è due volte <lb/>tanto quanto è la po&longs;&longs;anza di D, & di quella <lb/>che &longs;ta in H. </s> <s>Per laqual co&longs;a la po&longs;&longs;anza di L <lb/>viene ad e&longs;&longs;ere &longs;otto &longs;e&longs;quialtera del pe&longs;o G. </s> <s><lb/>Adunque il pe&longs;o G ver&longs;o la po&longs;&longs;anza di L è co­<lb/>me tre à due. </s> <s>cioè vna volta & meza. </s> <s>che bi&longs;o­<lb/>gnaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note256"></margin.target><emph type="italics"/>Per il corol lario della<emph.end type="italics"/> 5. <emph type="italics"/>di que&longs;io.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note257"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s>“Per laqual co&longs;a la po&longs;&longs;anza di L è &longs;otto &longs;e&longs;quialtera del pe&longs;o G. </s> <s>Hò detto, che il <lb/>&longs;opraparticolare è il &longs;econdo genere de'moltiplici, la prima &longs;petie del quale è <lb/>tre à due, che è &longs;e&longs;quialtera, cioè vna volta & meza. </s> <s>Hor chi fà comparatione <lb/>al contrario di due à tre na&longs;ce la &longs;otto &longs;e&longs;quialtera, hauendo forza quella voce <lb/>&longs;otto di paragonare la minore quantita con la maggiore. </s> <s>La po&longs;&longs;anza dunque di <lb/>L &longs;arà in proportione co'l pe&longs;o G come dueà tre, & in que&longs;ta gui&longs;a deue&longs;i in­<lb/>tendere &longs;empre tale vocabolo. </s> </p> <pb pagenum="87"/> <p type="main"> <s>Ma &longs;e la po&longs;&longs;anza che moue il pe&longs;o &longs;arà in L: Dico lo &longs;patio <lb/>della po&longs;&longs;anza e&longs;&longs;ere vna volta & meza tanto, quanto lo &longs;pa­<lb/>tio del pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Stando le co&longs;e ifle&longs;&longs;e, peruenga la girella <lb/>ABC fin ad MNO, & la girella <lb/>DEF fin à PQR; & H in S; <lb/>& il pe&longs;o G fin in T. </s> <s>Et perche la <lb/>corda HABCDEFK è eguale alla <lb/>corda SMNOPQRK e&longs;&longs;endo la <lb/>corda i&longs;te&longs;&longs;a; & le corde che &longs;ono d'in­<lb/>torno à mezi cerchi ABCMNO &longs;o <lb/>no tra loro eguali, & quelle, che &longs;ono <lb/>d'intorno alli mezi cerchi DEF PQR <lb/>&longs;imilmente &longs;ono tra loro eguali; tolte <lb/>via dunque le corde AS CP RK <lb/>communi, &longs;aranno le due CO MA e­<lb/>guali alle tre DP HS FR. ma l'v­<lb/>na, & l'altra di CO AM &longs;eparata­<lb/>mente è eguale allo &longs;patio della po&longs;&longs;an­<lb/>za mo&longs;&longs;a. </s> <s>Per laqual co&longs;a le due CO <lb/>MA in&longs;ieme &longs;aranno due volte tanto <lb/>quanto lo &longs;patio della po&longs;&longs;anza; & le <lb/>tre DP HS FR in&longs;ieme con &longs;imile <lb/>modo &longs;aranno tre volte tanto quanto <lb/>lo &longs;patio del pe&longs;o mo&longs;&longs;o. </s> <s>Ma la metà, <lb/>cioè lo &longs;patio della po&longs;&longs;anza mo&longs;&longs;a, al­<lb/>la terza parte, cioè allo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o, ha proportione tale quale è dal <lb/>doppio della metà al doppio del terzo, <lb/>cioè come il tutto à duo terzi, che è come <lb/>tre à due. </s> <s>Lo &longs;patio dunque della po&longs;&longs;an <lb/>za po&longs;ta in L è vna volta & meza tan <lb/>to quanto lo &longs;patio del pe&longs;o G mo&longs;&longs;o. </s> <s><lb/>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <figure></figure> <p type="head"> <s>PROPOSITIONE XXI.</s> </p> <p type="main"> <s>Se à tre girelle di due taglie, l vna delle quali &longs;ia &longs;o&longs;tenuta dalla <lb/>po&longs;&longs;anza di &longs;opra con vna &longs;ola girella, & l'altra con due girel­<lb/>le &longs;ia po&longs;ta di &longs;otto, & legata al pe&longs;o, &longs;arà inuolta d'intorno <lb/>la corda, con l'vno de' &longs;uoi capi legato in qualche luogo, & <lb/>l'altro legato nella taglia di &longs;opra; il pe&longs;o &longs;arà vna volta, & vn <lb/>terzo tanto quanto la po&longs;&longs;anza. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A legato alla taglia di <lb/>&longs;otto, laquale habbia due girelle, i <lb/>cui centri &longs;iano BC, & la taglia <lb/>di &longs;opra habbia la girella co'l centro <lb/>D; & &longs;ia la corda EFGHKL <lb/>MN riuolta d'intorno à tuttele gi­<lb/>relle, laquale &longs;ia legata in N, & <lb/>in E dalla taglia di &longs;opra; & &longs;ia <lb/>la po&longs;&longs;anza in O, che &longs;o&longs;tenga il pe <lb/>&longs;o A. </s> <s>Dico che il pe&longs;o è vna volta <lb/>& vn terzo tanto quanto è la po&longs;&longs;an <lb/>za. </s> <s>Et percioche cia&longs;cheduna delle<emph.end type="italics"/><lb/> <arrow.to.target n="note258"></arrow.to.target> <emph type="italics"/>corde NM HG EF KL &longs;o&longs;tie­<lb/>ne la quarta parte del pe&longs;o A; & <lb/>tutte in&longs;ieme &longs;o&longs;tengono tutto il pe­<lb/>&longs;o; le tre HG EF KL in&longs;ieme <lb/>&longs;o&longs;ierranno le tre parti del pe&longs;o A. </s> <s><lb/>Per laqual co&longs;a il pe&longs;o A ver&longs;o tut <lb/>te que&longs;te in&longs;ieme &longs;arà come quattro <lb/>à tre: & concio&longs;ia che la po&longs;&longs;anza di <lb/>O faccia il mede&longs;imo, che &longs;anno le <lb/>corde HG EF KL tutte in&longs;ie­<lb/>me; peroche le &longs;o&longs;tiene tutte; &longs;arà la <lb/>po&longs;&longs;anza di O eguale à le tre HG <lb/>EF KL in&longs;ieme; & perciò il pe&longs;o <lb/>A ver&longs;o la po&longs;&longs;anza di O &longs;arà co­<lb/>me quattro à tre, cioè vna volta, & <lb/>vn cerzo. </s> <s>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note258"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 1. <emph type="italics"/>co volario della<emph.end type="italics"/> V. <emph type="italics"/>di que&longs;to.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="88"/> <p type="main"> <s>Ma &longs;e in O &longs;arà la po&longs;&longs;an­<lb/>za che moua il pe&longs;o A. </s> <s><lb/>Dico lo &longs;patio cor&longs;o dal­<lb/>la po&longs;&longs;anza di O e&longs;&longs;ere <lb/>vna volta & vn terzo tan­<lb/>to quanto è lo &longs;patio del <lb/>pe&longs;o A mo&longs;&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Stando le co&longs;e mede&longs;ime, &longs;ia il centro <lb/>B mo&longs;&longs;o in P; & C fin in <expan abbr="q;">que</expan> <lb/>& D in R; & E in S nel­<lb/>l'i&longs;te&longs;&longs;o tempo: & &longs;iano per li cen­<lb/>tri condotte le linee ML<36>ZFG <lb/>TV HK X<foreign lang="greek">*u</foreign> egualmente di&longs;tan <lb/>ti, & dall' orizonte, & fra &longs;e &longs;te&longs;­<lb/>&longs;e: &longs;imilmente, come nella prece­<lb/>dente &longs;i dimo&longs;trerà, le tre corde <lb/>XH SE <foreign lang="greek">*u</foreign>K e&longs;&longs;ere eguali alle <lb/>quattro TG VF ZL <36>M. & <lb/>percioche le tre XH SE <foreign lang="greek">*u</foreign>K &longs;o­<lb/>no in&longs;ieme tre volte tanto quanto <lb/>lo &longs;patio della po&longs;&longs;anza: ma le <lb/>quattro TG VF ZL <36>M in­<lb/>&longs;ieme &longs;ono quattro volte <expan abbr="tāto">tanto</expan> quan <lb/>to lo &longs;patio del pe&longs;o mo&longs;&longs;o; &longs;arà lo <lb/>&longs;patio della po&longs;&longs;anza ver&longs;o lo &longs;pa­<lb/>tio del pe&longs;o, come la terza parte <lb/>alla quarta parte. </s> <s>Ma la terza <lb/>parte ver&longs;o la quarta parte è come <lb/>tre terzi à tre quarti, cioè come il <lb/>tutto ver&longs;o tre quarti, che è come <lb/>quattro ver&longs;o tre. </s> <s>Lo &longs;patio dun­<lb/>que della po&longs;&longs;anza allo &longs;patio del <lb/>pe&longs;o mo&longs;&longs;o hà proportione di vna <lb/>volta & vn terzo. </s> <s>che era damo­<lb/>&longs;trar&longs;i.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e la corda in E &longs;arà inuolta d'in <lb/>torno vn'altra girella, laqual cor-<emph.end type="italics"/></s> </p> <pb pagenum="89"/> <p type="head"> <s>PROPOSITIONE XXII.</s> </p> <p type="main"> <s>Se all'vna & l'altra di cia&longs;cuna girella delle due taglie, l'vna del­<lb/>le quali &longs;ia &longs;o&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra po&longs;ta di <lb/>&longs;otto, & legata al pe&longs;o, &longs;arà condotta d'intorno la corda; con <lb/>l'vno de &longs;uoi capi legato in qualche luogo, & l'altro attaccato <lb/>alla taglia di &longs;opra. </s> <s>&longs;arà la po&longs;&longs;anza vna volta & meza tanto <lb/>quanto il pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la girella ABC della taglia legata <lb/>al pe&longs;o D; & EFG la girella del­<lb/>la taglia di &longs;opra, il cui centro &longs;ia H; <lb/>&longs;ia dapoi la corda KABCEFGL ri­<lb/>uolta d'intorno alle girelle, & legata <lb/>in L & in K alla taglia di &longs;opra; & <lb/>&longs;ia in M la po&longs;&longs;anza, che &longs;o&longs;tiene il<emph.end type="italics"/> <arrow.to.target n="note259"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o D. </s> <s>Dico che la po&longs;&longs;anza è vna <lb/>volta & meza quanto è il pe&longs;o. </s> <s>Hor<emph.end type="italics"/> <arrow.to.target n="note260"></arrow.to.target><lb/><emph type="italics"/>percioche la po&longs;&longs;anza di E &longs;o&longs;tenente <lb/>il pe&longs;o D è la metà meno del pe&longs;o D;<emph.end type="italics"/> <arrow.to.target n="note261"></arrow.to.target><lb/><emph type="italics"/>& la po&longs;&longs;anza di H è due volte quan <lb/>to la po&longs;&longs;anza po&longs;ta in E; &longs;arà la po&longs;­<lb/>&longs;anza di H eguale al pe&longs;o D; & con <lb/>cio&longs;ia, che la po&longs;&longs;anza di K &longs;ia la me­<lb/>tà meno del pe&longs;o D; &longs;aranno ambe­<lb/>due le po&longs;&longs;anze in&longs;ieme po&longs;te in HK <lb/>vna volta & meza quanto il pe&longs;o </s> <s>/>e&longs;&longs;endo adunque la po&longs;&longs;anza di M egua­<lb/>le à due po&longs;&longs;anze in HK pre&longs;e in&longs;ie­<lb/>me, &longs;i come di &longs;opra è &longs;tato dichiarato; <lb/>&longs;arà la po&longs;&longs;anza di M vna volta & <lb/>meza quanto il pe&longs;o D. che bi&longs;ogna­<lb/>ua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note259"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note260"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note261"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 2. <emph type="italics"/>co rollario del la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e la po&longs;&longs;anza che moue il pe&longs;o &longs;arà in <lb/>M, &longs;i mo&longs;trerà &longs;imilmente, come nelle <lb/>precedenti, lo &longs;patio del pe&longs;o e&longs;&longs;ere vna <lb/>volta & meza tanto quanto lo &longs;patio <lb/>della po&longs;&longs;anza.<emph.end type="italics"/></s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Et &longs;e la corda in K &longs;arà inuolta d'interno ad vn'altra girella, il cui centro &longs;ia N; <lb/>laquale dapoi &longs;ta rilegata alla taglia di &longs;otto in O; & la po&longs;&longs;anza di M &longs;o&longs;ten­<lb/>ga il pe&longs;o D. </s> <s>Dico la proportione <lb/>della po&longs;&longs;anza al pe&longs;o e&longs;&longs;ere vna <lb/>volta, & vn terzo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Hor percioche la po&longs;&longs;anza di E che<emph.end type="italics"/><lb/> <arrow.to.target n="note262"></arrow.to.target> <emph type="italics"/>&longs;o&longs;tiene il pe&longs;o D conla corda EC <lb/>BAKPO è vn terzo di e&longs;&longs;o D,<emph.end type="italics"/><lb/> <arrow.to.target n="note263"></arrow.to.target> <emph type="italics"/>& la po&longs;&longs;anza di H è due volte <lb/>tanto quanto e&longs;&longs;o E; &longs;arà la po&longs;­<lb/>&longs;anza di H &longs;otto &longs;e&longs;quialtera al pe­<lb/>&longs;o D. & nel modo iste&longs;&longs;o, per­<lb/>cioche la po&longs;&longs;anza di O, laquale <lb/>è come &longs;e fo&longs;&longs;e nel centro della gi­<lb/>rella ABC è vn terzo del pe&longs;o<emph.end type="italics"/><lb/> <arrow.to.target n="note264"></arrow.to.target> <emph type="italics"/>D, & la po&longs;&longs;anza di N è due <lb/>volte tanto quanto è e&longs;&longs;o O. &longs;arà <lb/>parimente la po&longs;&longs;anza di N &longs;otto <lb/>&longs;e&longs;quialtera al pe&longs;o D. </s> <s>Per laqual <lb/>co&longs;a due po&longs;&longs;anze in&longs;ieme po&longs;te in <lb/>HN &longs;uperano il pe&longs;o D d'vna <lb/>terza parte, & &longs;ono ver&longs;o il detto <lb/>D in ragione di vna volta & vn <lb/>terzo. </s> <s>& concio&longs;ia, che la po&longs;&longs;an­<lb/>za di M &longs;ia eguale alle due po&longs;&longs;an <lb/>ze di HN pre&longs;e in&longs;ieme, &longs;upere­<lb/>ra mede&longs;imamente la detta po&longs;&longs;an­<lb/>za di M il pe&longs;o D di vn terzo. </s> <s><lb/>Adunque la proportione della po&longs;­<lb/>&longs;anza po&longs;ta in M ver&longs;o il pe&longs;o D <lb/>è vna volta, & vn terzo. </s> <s>che bi­<lb/>&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note262"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note263"></margin.target><emph type="italics"/>Dalla<emph.end type="italics"/> 15. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note264"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 3. <emph type="italics"/>&<emph.end type="italics"/> 15. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e la po&longs;&longs;anza mouente il pe&longs;o &longs;a­<lb/>rà in M, con modo &longs;imile proue­<lb/>ra&longs;&longs;i lo &longs;patio del pe&longs;o D e&longs;&longs;ere vna <lb/>volta & vn terzo tanto quanto la <lb/>po&longs;&longs;anza di M.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et &longs;e la corda in O &longs;arà inuolta d'in­<lb/>torno ad vn'altra girella, laquale dapoi &longs;ia legata alla taglia di &longs;opra; nell'i&longs;te&longs;&longs;o <lb/>modo dimo&longs;treremo la proportione della po&longs;&longs;anza M, che &longs;o&longs;tiene il pe&longs;o e&longs;&longs;ere <lb/>vna volta & vn quarto tanto quanto il pe&longs;o. </s> <s>& &longs;e in M &longs;arà la po&longs;&longs;anza che <lb/>moue, &longs;imilmente mo&longs;treraßilo &longs;patio del pe&longs;o e&longs;&longs;ere vna volta & vn quarto tan<emph.end type="italics"/> <pb pagenum="90"/><emph type="italics"/>to quanto lo &longs;patio della po&longs;&longs;anza. </s> <s>& co&longs;i procedendo in infinito ritrouereme <lb/>qual &longs;i voglia proportione &longs;opraparticolare della po&longs;&longs;anza al pe&longs;o, & &longs;empre <lb/>mostreremo la po&longs;&longs;anza, che &longs;o&longs;tiene il pe&longs;o co&longs;i e&longs;&longs;ere ver&longs;o il pe&longs;o, come lo &longs;pa­<lb/>tio del pe&longs;o allo &longs;patio della po&longs;&longs;anza, che moue il pe&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma il mouimento della leua EG è come &longs;e G &longs;o&longs;&longs;e il &longs;o&longs;tegno, e&longs;&longs;endo la corda legata <lb/>in L, & il pe&longs;o, come &longs;e fo&longs;&longs;e appiccato in E, & la po&longs;&longs;anza nel mezo. </s> <s>Ma <lb/>della leua CA il &longs;o&longs;tegno è A, il pe&longs;o nelmezo, & la po&longs;&longs;anza in C. & il <lb/>K è il &longs;o&longs;tegno della leua PK, il pe&longs;o in P, & la po&longs;&longs;anza nel mezo. </s> <s>Le qua­<lb/>li co&longs;e tutte &longs;i dimo&longs;treranno, come nelle precedenti.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE XXIII.</s> </p> <p type="main"> <s>Se all'vna, & l'altra delle due girelle di due taglie, l'vna dellequa <lb/>li &longs;ia &longs;o&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra po&longs;ta à ba&longs;&longs;o, <lb/>& legata al pe&longs;o, &longs;ia menata intorno la corda, legando am­<lb/>bidue li &longs;uoi capi in qualche luogo, non già nelle taglie; la <lb/>po&longs;&longs;anza &longs;arà eguale al pe&longs;o. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia la girella della taglia di &longs;o <lb/>pra ABC, il cui centro D; <lb/>& la girella della taglia le­<lb/>gata al pe&longs;o H &longs;ia EFG; il <lb/>cui centro K; & &longs;ia la cor <lb/>da LEFGABCM ri­<lb/>uolta d'intorno alle girelle <lb/>& legatain LM; & &longs;ia <lb/>in N la po&longs;&longs;anza che &longs;o­<lb/>&longs;tiene il pe&longs;o H. </s> <s>Dico che <lb/>la po&longs;&longs;anza di N è egua­<lb/>le al pe&longs;o H. </s> <s>Prenda&longs;i <lb/>il punto O douunque &longs;i <lb/>&longs;ia nella corda AG. </s> <s>Hor <lb/>percioche &longs;e la po&longs;&longs;anza, <lb/>che &longs;o&longs;tiene il pe&longs;o H fo&longs;<emph.end type="italics"/><lb/> <arrow.to.target n="note265"></arrow.to.target> <emph type="italics"/>&longs;e in O, &longs;arebbe la metà <lb/>meno del pe&longs;o H, & la<emph.end type="italics"/><lb/> <arrow.to.target n="note266"></arrow.to.target> <emph type="italics"/>po&longs;&longs;anza po&longs;ta in D è due <lb/>volte quanto è quella di O, <lb/>ouero (che è l'i&longs;te&longs;&longs;o) di N; <lb/>&longs;arà la po&longs;&longs;anza di N e­<lb/>guale al pe&longs;o H. che bi­<lb/>&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note265"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note266"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s>Et &longs;e in N &longs;arà la po&longs; <lb/>&longs;anza, che moue il <lb/>pe&longs;o. </s> <s>Dico, che lo <lb/>&longs;patio della po&longs;&longs;an­<lb/>za po&longs;ta in N è e­<lb/>guale allo &longs;patio del <lb/>pe&longs;o H mo&longs;&longs;o. </s> </p> <p type="main"> <s> <arrow.to.target n="note267"></arrow.to.target> <emph type="italics"/>Percio che lo &longs;patio del punto O mo&longs;&longs;o è due volte tanto quanto è lo &longs;patio sì del pe<emph.end type="italics"/><lb/> <arrow.to.target n="note268"></arrow.to.target> <emph type="italics"/>&longs;o H mo&longs;&longs;o, come della po&longs;&longs;anza N mo&longs;&longs;a; &longs;arà lo &longs;patio della po&longs;&longs;anza N <lb/>allo &longs;patio del pe&longs;o H eguale.<emph.end type="italics"/></s> </p> <pb pagenum="91"/> <p type="margin"> <s><margin.target id="note267"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note268"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="head"> <s>Altramente.</s> </p> <p type="main"> <s><emph type="italics"/>Stando le co&longs;e i&longs;te&longs;&longs;e. </s> <s>&longs;ia tra­<lb/>portato il centro della gi­<lb/>rella ABC fin à P; & <lb/>la girella habbia il &longs;ito in <lb/>QRS. </s> <s>Dapoi nell'i&longs;te&longs;&longs;o <lb/>tempo la girella EFG &longs;ia <lb/>in TVX, il cui centro <lb/>&longs;ia <foreign lang="greek">*u</foreign>, & il pe&longs;o &longs;ia per <lb/>uenuto in Z. &longs;iano tira­<lb/>te per i centri delle girel­<lb/>le le linee GETX AC <lb/>QS egualmente di&longs;tanti <lb/>dall' orizonte. </s> <s>& &longs;i come <lb/>nelle altre fu dimo&longs;trato, <lb/>le due corde AQ CS &longs;a <lb/>ranno eguali alle due cor­<lb/>de XG TE; ma AQ <lb/>CS in&longs;ieme &longs;ono due vol <lb/>te tanto quanto lo &longs;patio <lb/>della po&longs;&longs;anza mo&longs;&longs;a; & <lb/>le due XG TE in&longs;ieme <lb/>&longs;imilmente &longs;ono due vol­<lb/>te tanto quanto lo &longs;patio <lb/>del pe&longs;o; &longs;arà dunque lo <lb/>&longs;patio della po&longs;&longs;anza egua <lb/>le allo &longs;patio del pe&longs;o. </s> <s>che <lb/>bi&longs;ognaua mostrare.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e l'vna, & l'altra taglia haurà etiandio due girelle, i cui centri &longs;iano ABCD, <lb/>& la corda &longs;ia inuolta d'intorno à tutte, la quale &longs;ia rilegata in LM; &longs;imilmen­<lb/>te &longs;i mo&longs;trerà, che la po&longs;&longs;an­<lb/>za di N è eguale al pe&longs;o H. </s> <s><lb/>Peroche cia&longs;cuna po&longs;&longs;anza <lb/>po&longs;ta in EF &longs;o&longs;tenente il <lb/>pe&longs;o è vn quarto del pe&longs;o; & <lb/>le po&longs;&longs;anze di CD &longs;ono due <lb/>volte tanto quanto quelle, <lb/>che &longs;ono in EF; &longs;arà cia­<lb/>&longs;cuna po&longs;&longs;anza di CD la <lb/>metà del pe&longs;o H. </s> <s>Per la­<lb/>qual co&longs;a le po&longs;&longs;anze di CD <lb/>pre&longs;e in&longs;ieme &longs;aranno eguali <lb/>al pe&longs;o H. </s> <s>Et percioche la <lb/>po&longs;&longs;anza di N è eguale à <lb/>due po&longs;&longs;anze po&longs;te in CD; <lb/>&longs;arà la po&longs;&longs;anza di N egua <lb/>le al pe&longs;o H.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Et &longs;e la po&longs;&longs;anza che moue &longs;a­<lb/>rà in N, con modo &longs;imile <lb/>&longs;i mo&longs;trerà lo &longs;patio della po&longs; <lb/>&longs;anza e&longs;&longs;ere eguale allo &longs;pa­<lb/>tio del pe&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma &longs;e l'vna & l'altra taglia ha­<lb/>uerà tre, ò quattro, oue­<lb/>ro quante &longs;i voglia girelle, <lb/>&longs;empre &longs;i dimostrerà la po&longs;­<lb/>&longs;anza di N e&longs;&longs;ere eguale al <lb/>pe&longs;o H; & lo &longs;patio della <lb/>po&longs;&longs;anza mouente il pe&longs;o e&longs;­<lb/>&longs;ere eguale allo &longs;patio del pe <lb/>&longs;o mo&longs;&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma i mouimenti delle leue in <lb/>que&longs;ta maniera &longs;ono di&longs;po&longs;ti, <lb/>che il &longs;o&longs;tegno delle girelle <lb/>della taglia di &longs;opra, come <lb/>AC della figura preceden­<lb/>te è in C, il pe&longs;o appiccato in A, & la po&longs;&longs;anza nel mezo in D. ma le leue <lb/>delle girelle della taglia di &longs;otto co&longs;i &longs;i mouono, che di e&longs;&longs;o GE il &longs;o&longs;tegno &longs;ia E, <lb/>il pe&longs;o appiccato nel mezo, & la po&longs;&longs;anza in G.<emph.end type="italics"/></s> </p> <pb pagenum="92"/> <p type="head"> <s>PROPOSITIONE XXIIII.</s> </p> <p type="main"> <s>Se à tre girelle di due taglie, l'vna delle quali, che habbia vna gi <lb/>rella &longs;olamente &longs;ia &longs;o&longs;tenuta di &longs;opra dalla po&longs;&longs;anza, & l'altra <lb/>po&longs;ta di &longs;otto con due girelle, & legata al pe&longs;o, &longs;arà girata in­<lb/>torno la corda: e&longs;&longs;endo li due &longs;uoi capi legat in qualche luo <lb/>go, ma non già nella taglia di &longs;opra: il pe&longs;o &longs;arà il doppio del <lb/>la po&longs;&longs;anza. </s> </p> <p type="main"> <s><emph type="italics"/>Siano AB i centri delle girelle della <lb/>taglia legata al pe&longs;o C: & il D <lb/>&longs;ia il centro della girella di &longs;opra; <lb/>&longs;ia dapoi la corda riuolta d'intorno <lb/>à tutte le girelle, & rilegata in EF; <lb/>& &longs;ia in G la po&longs;&longs;anza, che &longs;o­<lb/>&longs;tiene il pe&longs;o C. </s> <s>Dico, che il pe&longs;o <lb/>C è due volte tanto quanto la <lb/>po&longs;&longs;anza G. </s> <s>Hor percioche &longs;e in <lb/>HK fo&longs;&longs;ero due po&longs;&longs;anze, che &longs;o­<lb/>&longs;tene&longs;&longs;ero il pe&longs;o con due corde ri­<lb/>uolte d'intorno alle girelle &longs;olamen <lb/>te della taglia di &longs;otto, &longs;arebbe per <lb/>certo l'vna & l'altra po&longs;&longs;anza po-<emph.end type="italics"/> <arrow.to.target n="note269"></arrow.to.target><lb/><emph type="italics"/>&longs;ta in KH vn quarto del pe&longs;o C; <lb/>Mala po&longs;&longs;anza di G è eguale alle <lb/>po&longs;&longs;anze di HK pre&longs;e in&longs;ieme:<emph.end type="italics"/> <arrow.to.target n="note270"></arrow.to.target><lb/><emph type="italics"/>percioche è due volte tanto quan­<lb/>to cia&longs;cuna delle po&longs;&longs;anze di H, <lb/>& K; &longs;arà la po&longs;&longs;anza di G la <lb/>metà del pe&longs;o C. il pe&longs;o dunque <lb/>&longs;arà il doppio della po&longs;&longs;anza. </s> <s>che <lb/>bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note269"></margin.target><emph type="italics"/>Dalla<emph.end type="italics"/> 7. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note270"></margin.target><emph type="italics"/>Dalla<emph.end type="italics"/> 15. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s>Et &longs;e in G &longs;arà la po&longs;&longs;anza <lb/>mouente il pe&longs;o. </s> <s>Dico <lb/>che lo &longs;patio della po&longs;&longs;an <lb/>za è il doppio dello &longs;pa­<lb/>tio del pe&longs;o. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Stando le co&longs;e i&longs;te&longs;&longs;e. </s> <s>&longs;iano mo&longs;&longs;e le gi <lb/>relle; &longs;i dimo&longs;trerà &longs;imilmente am­<lb/>bedue quelle corde LM NO e&longs;­<lb/>&longs;ere eguali alle quattro PQ RS <lb/>TV X<foreign lang="greek">*u. </foreign></s> <s>Ma LM NO in­<lb/>&longs;ieme &longs;ono il doppio dello &longs;patio <lb/>della po&longs;&longs;anza di G mo&longs;&longs;a; & <lb/>le quattro PQ RS TV X<foreign lang="greek">*u</foreign><lb/>in&longs;ieme &longs;ono quattro volte tanto <lb/>quanto lo &longs;patio del pe&longs;o mo&longs;&longs;o. </s> <s><lb/>Lo &longs;patio dunque della po&longs;&longs;anza <lb/>ver&longs;o lo &longs;patio del pe&longs;o è come la <lb/>metà ad vn quarto. </s> <s>Sarà dunque <lb/>lo &longs;patio della po&longs;&longs;anza allo &longs;patio <lb/>del pe&longs;o il doppiò.<emph.end type="italics"/></s> </p> <pb pagenum="93"/> <p type="main"> <s><emph type="italics"/>Di qui egli è da con&longs;iderare <lb/>in che modo &longs;i faccia il mo <lb/>uimento; percioche e&longs;&longs;en­<lb/>do legata la corda in F, <lb/>la leua NO nella prima <lb/>figura haurà il &longs;o&longs;tegno in <lb/>O, il pe&longs;o nel mezo, & la <lb/>po&longs;&longs;anza in N. &longs;rmil. </s> <s><lb/>mente percioche la corda <lb/>è rilegata in E, la leua <lb/>PQ haurà il &longs;o&longs;tegno in <lb/>P, & il pe&longs;o ne, nezo, <lb/>& la po&longs;&longs;anza in <expan abbr="q.">que</expan> On <lb/>de le parti delle girelle di <lb/>N & Q &longs;i moueranno <lb/>in sù; adunque le girelle <lb/>&longs;i moueranno non ad vna <lb/>parte, ma in contrarie par <lb/>ti, cioè vna alla de&longs;tra, & <lb/>l'altra alla &longs;ini&longs;tra. </s> <s>& <lb/>percioche le po&longs;&longs;anze di <lb/>NQ &longs;on<gap/>le i&longs;te&longs;&longs;e, che <lb/>&longs;ono in LM; le po&longs;&longs;an­<lb/>ze dunque di LM e&longs;&longs;en­<lb/>do eguali &longs;i moueranno <lb/>in sù. </s> <s>La leua dunque <lb/>LM non &longs;i mouerà in <lb/>niuna delle parti. </s> <s>Per la <lb/>qual co&longs;a no anche la gi­<lb/>rella &longs;i girerà intorno. </s> <s><lb/>Co&longs;i LM &longs;arà come bi­<lb/>lancia, il cni centro D, <lb/>& li pe&longs;i appiccati in LM <lb/>&longs;aranno eguali alla quar-<emph.end type="italics"/><lb/> <arrow.to.target n="fig105"></arrow.to.target><lb/><emph type="italics"/>ta parte del pe&longs;o C; peroche cia&longs;cheduna corda in LN MQ &longs;o&longs;tiene la quar <lb/>ta parte del pe&longs;o C; &longs;i mouerà dunque tuttala girella, il cui centro è D in sù, <lb/>ma non già voltera&longs;&longs;i intorno.<emph.end type="italics"/></s> </p> <pb/> <figure id="fig105"></figure> <p type="main"> <s><emph type="italics"/>Et &longs;e la corda po&longs;ta in F &longs;iriuolgerà <lb/>d'intorno à due altre girelle, i cui <lb/>centri &longs;iano HK l'aquale dapoi &longs;ia <lb/>rilegata in L; &longs;arà la proportione <lb/>del pe&longs;o alla po&longs;&longs;anza vna volta & <lb/>meza.<emph.end type="italics"/></s> </p> <p type="main"> <s> <arrow.to.target n="note271"></arrow.to.target> <emph type="italics"/>Percioche &longs;e fo&longs;&longs;ero quattro po&longs;&longs;anze in <lb/>MNOI, cia&longs;cheduna di loro &longs;areb <lb/>be vn &longs;e&longs;to del pe&longs;o C. </s> <s>Per laqual <lb/>co&longs;a quattro po&longs;&longs;anze in&longs;ieme in MN <lb/>OI &longs;aranno quattro &longs;e&longs;ti del pe&longs;o C. <lb/>& percioche due po&longs;&longs;anze in&longs;ieme po <lb/>&longs;te in HD &longs;ono eguali à quattro po&longs; <lb/>&longs;anze po&longs;te in MNOI; & la po&longs;­<lb/>&longs;anza di G è eguale alle po&longs;&longs;anze di <lb/>DH; &longs;arà la po&longs;&longs;anza di G egua­<lb/>le à quattro po&longs;&longs;anze in&longs;ieme po&longs;te <lb/>in MNOI; & perciò &longs;arà quat­<lb/>tro &longs;e&longs;ti del pe&longs;o C. </s> <s>La proportio­<lb/>ne dunque del pe&longs;o C alla po&longs;&longs;anza <lb/>di G è vna volta & meza.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note271"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Et &longs;e in G &longs;arà la po&longs;&longs;anza, che moue, <lb/>con modo &longs;imile &longs;i mo&longs;trerà lo &longs;patio <lb/>della po&longs;&longs;anza e&longs;&longs;ere vna volta & <lb/>meza tanto quanto lo &longs;patio del pe&longs;o.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et &longs;e la corda di L &longs;arà dauantaggio <lb/>riuolta d'intorno due altre girelle, &longs;i­<lb/>milmente &longs;i dimo &longs;trerà la proportio­<lb/>ne del pe&longs;o alla po&longs;&longs;anza e&longs;&longs;ere vna <lb/>volta, & vn terzo. </s> <s>Che &longs;e in G <lb/>&longs;arà la po&longs;&longs;anza che moue, &longs;i mo&longs;tre­<lb/>rà lo &longs;patio della po&longs;&longs;anza e&longs;&longs;ere vna <lb/>volta, & vn terzo quanto lo &longs;patio <lb/>del pe&longs;o, & co&longs;i di mano in mano <lb/>procedendo in infinito ritroueremo <lb/>qual &longs;i voglia proportione &longs;oprapar­<lb/>ticolare del pe&longs;o alla po&longs;&longs;anza. </s> <s>& <lb/>&longs;empre ritroueremo co&longs;i e&longs;&longs;ere il pe&longs;o <lb/>ver&longs;o la po&longs;&longs;anza che lo &longs;o&longs;tiene, co­<lb/>me lo &longs;patio della po&longs;&longs;anza che moue <lb/>allo &longs;patio del pe&longs;o mo&longs;&longs;o dalla po&longs;­<lb/>&longs;anza.<emph.end type="italics"/></s> </p> <pb pagenum="94"/> <p type="main"> <s><emph type="italics"/>Il mouimento delle leue &longs;i fa in que&longs;to modo, la leua <foreign lang="greek">*u</foreign> Z, e&longs;&longs;endo la corda legata in <lb/>E ha il &longs;o&longs;tegno in <foreign lang="greek">*u</foreign>, il pe&longs;o attaccato in B nel mezo, & la po&longs;&longs;anza in Z. <lb/>& la leua PQ hail &longs;ostegno in P, la po&longs;&longs;anza nel mezo, & il pe&longs;o in <expan abbr="q.">que</expan> <lb/>Percioche bi&longs;ogna, che le girelle, i cui centri &longs;ono BD, &longs;i mouano nella parte <lb/>i&longs;te&longs;&longs;a, cioè che QZ &longs;i mouano all'insu. </s> <s>& percioche la corda è rilegata in L, <lb/>&longs;ara il T il &longs;o&longs;iegno della leua ST, che hail pe&longs;o nel mezo, & la po&longs;&longs;anza in <lb/>S; & percioche S &longs;i moue all'insù, è co&longs;a nece&longs;&longs;aria, che R anchora &longs;i moua <lb/>all'insù; & però, F &longs;arà il &longs;o&longs;tegno della leua FR, & il pe&longs;o &longs;arà in R, & <lb/>la po&longs;&longs;anza nel mezo. </s> <s>Le girelle dunque, i cui centri &longs;ono HK &longs;i mouono in <lb/>parti contrarie di quelle, le quali hanno i centri BD; Per laqual co&longs;a le parti del­<lb/>le girelle PF nelle girelle inchineranno al ba&longs;&longs;o, cioè ver&longs;o XV. </s> <s>La leua dun <lb/>que VX non &longs;i mouerà nè in vna, nè in altra parte, mouendo&longs;i P & F al <lb/>ba&longs;&longs;o; & VX &longs;arà come leua, nel cui mezo &longs;ia appiccato il pe&longs;o, & in VX <lb/>due po&longs;&longs;anze eguali alla &longs;e&longs;ta parte del pe&longs;o C. </s> <s>Percioche le po&longs;&longs;anze di MO, <lb/>cioè le corde PV FX &longs;o&longs;tengono la &longs;e&longs;ta parte del pe&longs;o C. </s> <s>Adunque tutta <lb/>la girella, il cui centro è A &longs;i mouerà in sù in&longs;ieme con la taglia, ma non già &longs;i <lb/>volgerà intorno.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE XXV.</s> </p> <p type="main"> <s>Se à tre girelle di due taglie, l'vna delle quali habbia due girel­<lb/>le, & &longs;ia tenuta di &longs;opra dalla po&longs;&longs;anza; & l'altra habbia vna &longs;o <lb/>la girella, & &longs;ia po&longs;ta di &longs;otto, & legata al pe&longs;o, &longs;arà inuolta in <lb/>torno la corda: e&longs;&longs;endo legato l'vn & l'altro de'&longs;uoi capi in <lb/>qualche luogo, ma non già nella taglia di &longs;otto. </s> <s>La pos&longs;anza <lb/>&longs;arà due volte tanto quanto il pe&longs;o. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia il pe&longs;o A legato alla taglia di &longs;otto, laquale habbia la girella &longs;uä co'l centro B; <lb/>ma la taglia di &longs;opra habbia due girelle, i cui centri &longs;iano CD, & &longs;ia la corda <lb/>inuolta d'intorno à tutte le girelle, & rilegata in EF; & la po&longs;&longs;anza che fo&longs;tie­<lb/>ne il pe&longs;o &longs;ia in G. </s> <s>Di­<lb/>co la po&longs;&longs;anza di G e&longs;&longs;e­<lb/>re due volte tanto quanto<emph.end type="italics"/><lb/> <arrow.to.target n="note272"></arrow.to.target> <emph type="italics"/>il pe&longs;o. </s> <s>A. </s> <s>Percioche &longs;e <lb/>in HK &longs;o&longs;&longs;ero due po&longs;&longs;an <lb/>ze, che &longs;o&longs;tene&longs;&longs;ero il pe­<lb/>&longs;o, l'vna & l'altra &longs;areb-<emph.end type="italics"/><lb/> <arrow.to.target n="note273"></arrow.to.target> <emph type="italics"/>be la metà del pe&longs;o A: <lb/>ma la po&longs;&longs;anza di D è due <lb/>volte tanto quanto la po&longs;­<lb/>&longs;anza di H, & la po&longs;&longs;an <lb/>za di C è due volte tanto <lb/>quanto la po&longs;&longs;anza di K; <lb/>Per laqual co&longs;a due po&longs;­<lb/>&longs;anze in&longs;ieme po&longs;te in CD <lb/>&longs;aranno il doppio di ambe­<lb/>due le po&longs;&longs;anze di HK pre <lb/>&longs;e in&longs;ieme. </s> <s>Male po&longs;&longs;an­<lb/>ze di HK &longs;ono eguali al <lb/>pe&longs;o A & le po&longs;&longs;anze di <lb/>CD &longs;ono etiandio eguali <lb/>ad e&longs;&longs;a po&longs;&longs;anza di G; la <lb/>po&longs;&longs;anza dunque di G &longs;a­<lb/>rà il doppio del pe&longs;o A, <lb/>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note272"></margin.target><emph type="italics"/>Per il<emph.end type="italics"/> 2. <emph type="italics"/>co rollario del la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note273"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e in G &longs;arà la po&longs;&longs;anza <lb/>mouente il pe&longs;o, &longs;imilmen <lb/>te &longs;i mo&longs;trerà, come nella <lb/>precedente lo &longs;patio del <lb/>pe&longs;o e&longs;&longs;ere il doppio dello <lb/>&longs;patio della po&longs;&longs;anza.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Qui parmente è da con&longs;ide­<lb/>rare, che la leua PQ non &longs;i moue, peroche la leua LM hà il &longs;o&longs;tegno in L, la <lb/>po&longs;&longs;anza nel mezo, & il pe&longs;o in M. </s> <s>Mala leua NO hà il &longs;o&longs;tegno in O, la po&longs; <lb/>&longs;anza nel mezo, & il pe&longs;o in N. </s> <s>Per laqual co&longs;a M, & N &longs;i moueranno all'in <lb/>sù. </s> <s>Le girelle dunque, lequali hanno i centri CD &longs;i mouono in parti contrarie. </s> <s><lb/>Onde la leua PQ non &longs;i mouerà nè all'vna, nè all'altra parte; & &longs;arà come &longs;e <lb/>fo&longs;&longs;e appiccato il pe&longs;o nel mezo, & in PQ due po&longs;&longs;anze fu&longs;&longs;ero eguali alla metà <lb/>del pe&longs;o A. </s> <s>Peroche l'vna & l'altra po&longs;&longs;anza di HK è la metà del pe&longs;o A.<emph.end type="italics"/> <pb pagenum="95"/><emph type="italics"/>Tutta la girella dunque il cui centro è B &longs;i mouerà all'insù, ma non già &longs;i volge­<lb/>rà intorno.<emph.end type="italics"/></s> </p> <p type="main"> <s>Et &longs;e la corda di F &longs;i volge&longs;&longs;e <lb/>ancora d'intorno à due al­<lb/>tre girelle, i cui centri &longs;o&longs;­<lb/>&longs;ero HK, laqual corda <lb/>poi &longs;ia legata in L; &longs;arà la <lb/>proportione della po&longs;&longs;an­<lb/>za po&longs;ta in G vna volta <lb/>& meza quanto il pe&longs;o A. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Percioche &longs;e in MN OP fo&longs;&longs;ero quat <lb/>tro po&longs;&longs;anze &longs;o&longs;tenenti il pe&longs;o, cia-<emph.end type="italics"/> <arrow.to.target n="note274"></arrow.to.target><lb/><emph type="italics"/>&longs;cheduna di loro &longs;arebbe il quarto del <lb/>pe&longs;o A: ma concio&longs;ia che la po&longs;&longs;an<emph.end type="italics"/> <arrow.to.target n="note275"></arrow.to.target><lb/><emph type="italics"/>za di K &longs;ia il doppio della po&longs;&longs;anza <lb/>di N; &longs;arà la po&longs;&longs;anza di K vn <lb/>quarto del pe&longs;o A. & percioche <lb/>la po&longs;&longs;anza po&longs;ta in D è eguale al <lb/>le due po&longs;&longs;anze MO; &longs;arà ancho­<lb/>rala po&longs;&longs;anza di D vn quarto del <lb/>pe&longs;o A. </s> <s>Et di più e&longs;&longs;endo la po&longs;­<lb/>&longs;anza di C vn quarto della po&longs;&longs;an­<lb/>za di P, &longs;arà &longs;imilmente la po&longs;&longs;an <lb/>za di C vn quarto del pe&longs;o A. </s> <s><lb/>Tre po&longs;&longs;anze dunque po&longs;te in CDK <lb/>&longs;ono eguali à tre metà del pe&longs;o A. </s> <s><lb/>Ma percioche la po&longs;&longs;anza di G è <lb/>eguale alle po&longs;&longs;anze di CDK, &longs;a­<lb/>rà la po&longs;&longs;anza di G eguale alle tre <lb/>metà del pe&longs;o A. </s> <s>La proportio­<lb/>ne dunque della po&longs;&longs;anza al pe&longs;o è <lb/>vna volta, & meza.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note274"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 7. <emph type="italics"/><gap/> questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note275"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/><gap/> questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Che &longs;e in G &longs;arà la po&longs;&longs;anza, che mo­<lb/>ue, &longs;arà lo &longs;patio del pe&longs;o vna volta <lb/>& meza tanto quanto lo &longs;patio del­<lb/>la po&longs;&longs;anza.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et &longs;e la eorda in L &longs;arà inuolta da<gap/>an- <pb/>tag gio d'intorno à due altre girelle, <lb/>&longs;imilmente &longs;i mostrerà la proportio­<lb/>ne della po&longs;&longs;anza al pe&longs;o e&longs;&longs;ere vna <lb/>volta & vn terzo. </s> <s>& co&longs;i in infini <lb/>to ritroueremo tutte le proportioni <lb/>&longs;opraparticolari della po&longs;&longs;anza al pe <lb/>&longs;o. </s> <s>& mo&longs;treremo la po&longs;&longs;anza che <lb/>&longs;o&longs;tiene il pe&longs;o e&longs;&longs;ere co&longs;i ver&longs;o il pe­<lb/>&longs;o, come lo &longs;patio d l pe&longs;o mo&longs;&longs;o <lb/>allo &longs;patio della po&longs;&longs;anza che mo­<lb/>ue il pe&longs;o.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Il mouimento delle leue &longs;i farà in que­<lb/>&longs;to modo, cioè il Q &longs;arà il &longs;o&longs;tegn<gap/><lb/>della leua QR, la po&longs;&longs;anza nel me <lb/>zo, il pe&longs;o in R; & della leua Z <36><lb/>il &longs;o&longs;tegno &longs;arà il Z, il pe&longs;o nel <lb/>mezo, & la po&longs;&longs;anza in <36>. </s> <s>&longs;imil­<lb/>mente lo X &longs;arà il &longs;o&longs;tegno della le <lb/>ua VX, la po&longs;&longs;anza nel mezo, & <lb/>il pe&longs;o in V. & percioche lo V <lb/>&longs;i moue all'insù, &longs;i mouerà all in sù lo <lb/><foreign lang="greek">*u</foreign> ancora, & della leua <foreign lang="greek">*u</foreign> F il &longs;o­<lb/>&longs;tegno &longs;arà F. </s> <s>Per laqual co&longs;a F <lb/>& Z nelle girelle &longs;i moueranno in <lb/>giù. </s> <s>& perciò la leua ST non &longs;i <lb/>mouerà nè in vna, nè in altna par­<lb/>te; & ST &longs;arà come bilancia, il <lb/>cui centro &longs;arà D, & i pe&longs;i po&longs;ti <lb/>in ST &longs;aranno eguali alla quarta <lb/>parte del pe&longs;o A. </s> <s>Peroche cia&longs;cu­<lb/>na corda SZ TF &longs;o&longs;tiene la quar­<lb/>ta parte del pe&longs;o A. </s> <s>La girellá <lb/>dunque del centro D &longs;i monerà al­<lb/>l'insù, ma non &longs;i volgerà intorno.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Fin qui, &longs;ono &longs;tate dichiarate le propor <lb/>tioni molteplici, & &longs;otto molteplici <lb/>che ha il pe&longs;o alla po&longs;&longs;anza; & da­<lb/>poi le proportioni &longs;opraparticolari, <lb/>& &longs;otto &longs;opraparticolari. </s> <s>Hora re­<lb/>&longs;ta, che &longs;i manife&longs;tino le proportio-<emph.end type="italics"/> <pb pagenum="96"/><emph type="italics"/>ni tra il pe&longs;o, & la po&longs;&longs;anza &longs;oprapartienti, & molteplici &longs;opraparticolari, & <lb/>molteplici &longs;oprapartienti.<emph.end type="italics"/></s> </p> <p type="main"> <s>Et dapoi le &longs;opraparticolari, & le &longs;otto &longs;opraparticolari furono dichiarate. </s> <s>“Dal co <lb/>no&longs;cimento del &longs;opraparticolare &longs;i intende ageuolmente il &longs;otto &longs;opraparticolare <lb/>che gli è oppo&longs;to; pero che paragonando come è detto il 3. co'l 2. na&longs;ce il &longs;o­<lb/>praparticolare, & per lo contrario il 2. co'l 3. &longs;i produce il &longs;otto &longs;opraparticolare <lb/>per la forza di quella voce &longs;otto. </s> </p> <p type="main"> <s>Hora re&longs;ta &c. </s> <s>“Qui propone di trattare delle proportioni, che il pe&longs;o hà con la po&longs; <lb/>&longs;anza nel genere &longs;oprapartiente, & nel genere compo&longs;to del molteplice &longs;oprapar <lb/>ticolare, & del molteplice &longs;oprapa<gap/>tiente. </s> <s>il genere &longs;o prapartiente è diuer&longs;o dal <lb/>&longs;opraparticolare, che doue nel &longs;opraparticolare vna quantità contiene l'altra vna <lb/>ò più volte, & più parte, che può interamente numerare & l'vna, & l'altra: nel <lb/>&longs;oprapartiente contiene vna, ò più<gap/>volte, & dauantaggio parte che non le puo­<lb/>te numerare, & mi&longs;urare perfettamente, come il cinque contiene il 3. vna volta, <lb/>& piu parte di e&longs;&longs;o, che è il 2. il quale non è mi&longs;ura commune di ambidue loro, <lb/>& &longs;i denomina &longs;oprabipartiente terze, pero che contiene vna volta, & piu due <lb/>terze parti del contenuto. </s> </p> <p type="main"> <s>Segue poi. </s> <s>Et le molteplici &longs;opraparti<gap/>olari, che hò di &longs;opra mo&longs;trato. </s> <s>“Componen <lb/>do due generi in&longs;ieme il molteplice, & il &longs;o praparticolare na&longs;ce que&longs;to moltepli­<lb/>ce &longs;o praparticolare, nelquale vna quantità contiene l'altra molte volte, & più par <lb/>te di e&longs;&longs;a, che è mi&longs;ura commune di<gap/>ambedue. </s> <s>La primiera &longs;ua &longs;petie è il 5. pa­<lb/>ragonato co'l due, che lo contiene due volte, & piu la metà di lui, cioè vno, mi­<lb/>&longs;ura di ambedue. </s> <s>Chiama&longs;i que&longs;ta proportione doppia &longs;e&longs;quialtera. </s> <s>Mettendo <lb/>parimente in&longs;ieme il genere molteplice co'l &longs;oprapartiente, &longs;i fa il molteplice &longs;o­<lb/>prapartiente, il quale è differente dal &longs;o pradetto per ri&longs;petto che in lui la maggior <lb/>quantità contiene la minore molte volte, & piu parte di e&longs;&longs;a, che non puote e&longs;&longs;e­<lb/>re loro mi&longs;ura commune; la prima &longs;petie del qual genere è come 8. à 3. peroche <lb/>l'otto contiene il 3. due volte, & piu parte di e&longs;&longs;o 3. cioè 2. che non gli puo mi&longs;u <lb/>rare ambidue, concio&longs;ia che il 2. non puo mi&longs;urare il 3. come fà l'otto per e&longs;&longs;ere <lb/>que&longs;ti due numeri 8. & 3. tra &longs;e primi. </s> <s>& chiama&longs;i proportione doppia &longs;oprabi­<lb/>partiente. </s> <s>Vuole dunque l'autore andar inue&longs;tigando le proportioni fra il pe&longs;o, <lb/>& la po&longs;&longs;anza ne i predetti generi an cora, come hà fatto ne gli altri. </s> </p> <p type="main"> <s>Da que&longs;te poche co&longs;e, lequali hò qui narrato per ageuolare l'intédimento de i voca­<lb/>boli pertinenti alle proportioni po&longs;te da l'autore, &longs;i potrà facilmente con qual­<lb/>che &longs;tudio comprendere tutta la &longs;omma delle vltime dimo &longs;trationi della taglia, <lb/>nelle quali &longs;ono que&longs;ti vocaboli di proportioni, quantunque in ogni loco qua&longs;i <lb/>con gli e&longs;&longs;empi &longs;tes&longs;i de' numeri &longs;iano dall'autore manife&longs;tate. </s> </p> <p type="head"> <s>PROPOSITIONE XXVI.</s> </p> <p type="head"> <s>PROBLEMA.</s> </p> <p type="main"> <s>Se vogliamo trouare la proportione &longs;oprapartiente, come &longs;e la <lb/>proportione, laquale hà il pe&longs;o alla po&longs;&longs;anza che &longs;o&longs;tiene il pe <lb/>&longs;o &longs;arà &longs;oprabipartiente, come il cin que à tre. </s> </p> <p type="main"> <s> <arrow.to.target n="note276"></arrow.to.target> <emph type="italics"/>Ponga&longs;i la po&longs;&longs;anza in A, the &longs;o­<lb/>&longs;tenga il pe&longs;o B, & il pe&longs;o B <lb/>habbia proportione alla po&longs;&longs;an­<lb/>za A, come cinque ad vno; <lb/>cioè &longs;ia la po&longs;&longs;anza di A vn quin <lb/>to del pe&longs;o B: dapoiriuolgendo <lb/>la corda i&longs;te&longs;&longs;a d'intorno ad altre<emph.end type="italics"/><lb/> <arrow.to.target n="note277"></arrow.to.target> <emph type="italics"/>girelle, ritroui&longs;i la po&longs;&longs;anza di C, <lb/>laquale &longs;ia tre volte tanto quan­<lb/>to la po&longs;&longs;anza di A. </s> <s>Et percio <lb/>che il pe&longs;o B alla po&longs;&longs;anza po­<lb/>&longs;ta in A è come cinque ad vno; <lb/>& la po&longs;&longs;anza di A alla po&longs;&longs;an <lb/>za di C è come vno ver&longs;o tre, &longs;a <lb/>rà il pe&longs;o B ver&longs;o la po&longs;&longs;anza <lb/>di C come cinque à tre, cioè &longs;o­<lb/>prabipartiente.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note276"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note277"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Et à que&longs;to modo tutte le proportio­<lb/>ni &longs;oprapartienti del pe&longs;o alla po&longs; <lb/>&longs;anza &longs;i troueranno; come &longs;e la <lb/>proportione &longs;opratrepartiente vor <lb/>rà alcuno trouare, proceda con <lb/>l'ordine iste&longs;&longs;o: cioè &longs;accia&longs;i che la <lb/>po&longs;&longs;anza di A &longs;ostenente il pe­<lb/>&longs;o B &longs;ia vn &longs;ettimo del pe&longs;o B; <lb/>Dapoi &longs;i faccia, che la po&longs;&longs;anza <lb/>di C &longs;ia quattro volte tanto quan <lb/>to è quella di A; &longs;arà il pe&longs;o B <lb/>ver&longs;o la po&longs;&longs;anza di C, come &longs;et <lb/>te à quattro; cioè &longs;opratrepar­<lb/>tiente.<emph.end type="italics"/></s> </p> <p type="main"> <s>Ma &longs;e in C &longs;arà la po&longs;&longs;an­<lb/>za mouente il pe&longs;o, &longs;arà <lb/>lo &longs;patio della po&longs;&longs;anza <lb/>&longs;oprabipartiente allo &longs;pa <lb/>tio del pe&longs;o. </s> </p> <p type="main"> <s> <arrow.to.target n="note278"></arrow.to.target> <emph type="italics"/>Per cioche lo &longs;patio della po&longs;&longs;anza <lb/>po&longs;ta in C è la terza parte della <lb/>&longs;patio della po&longs;&longs;anza po&longs;ta in A,<emph.end type="italics"/> <pb pagenum="97"/><emph type="italics"/>cioè, che co&longs;i &longs;ono tra loro, come il cinque al quindici: & lo &longs;patio della po&longs;&longs;anza<emph.end type="italics"/> <arrow.to.target n="note279"></arrow.to.target><lb/><emph type="italics"/>di A è cinque volte tanto quanto lo &longs;patio del pe&longs;o B, cioè come quindici à tre. <lb/>&longs;arà dunque lo &longs;patio della po&longs;&longs;anza posta in C ver&longs;o lo &longs;patio del pe&longs;o B come <lb/>cinque à tre; cioè &longs;oprabipartiente: & &longs;empre dimostreremo, co&longs;i e&longs;&longs;ere lo &longs;patio <lb/>della po&longs;&longs;anza che moue allo &longs;patio del pe&longs;o; come il pe&longs;o alla po&longs;&longs;anza che lo &longs;o­<lb/>&longs;tiene.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note278"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note279"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 14. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et con ragione del tutto &longs;imile ritroueremo la proportione &longs;oprapartiente della po&longs;&longs;an <lb/>za al pe&longs;o. </s> <s>Peroche &longs;e C fo&longs;&longs;e di &longs;otto, & in e&longs;&longs;o fo&longs;&longs;e appiccato il pe&longs;o; & il <lb/>B di &longs;opra, nelquale fo&longs;&longs;e la po&longs;&longs;anza che in C &longs;o&longs;tiene il pe&longs;o, &longs;arebbe la po&longs;­<lb/>&longs;anza di B &longs;oprabipartiente al pe&longs;o appiccato in C: e&longs;&longs;endo il B allo A come<emph.end type="italics"/> <arrow.to.target n="note280"></arrow.to.target><lb/><emph type="italics"/>cinque ad vno; ma A al C come l'vno al tre.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note280"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 18. <emph type="italics"/>& per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s>Ma &longs;e vorremo trouare la proportione molteplice &longs;oprapartico­<lb/>lare; come &longs;e la proportione, laquale ha il pe&longs;o alla po&longs;&longs;anza, <lb/>che lo &longs;o&longs;tiene &longs;ia doppia &longs;e&longs;quialtera, come cinque à due. </s> </p> <p type="main"> <s><emph type="italics"/>Nell'i&longs;te&longs;&longs;o modo, co'l quale ritrouiamo le &longs;oprapartienti, ritroueremo ancora tutte que <lb/>ste molteplici &longs;opraparticolari. </s> <s>Come faccia&longs;i il pe&longs;o po&longs;to in B alla po&longs;&longs;anza di A,<emph.end type="italics"/> <arrow.to.target n="note281"></arrow.to.target><lb/><emph type="italics"/>come il cinque all'vno; & la po&longs;&longs;anza di C alla po&longs;&longs;anza di A come il due all'vno; <lb/>co&longs;a che &longs;i farà, &longs;ela corda &longs;arà rilegata in D, ouero in E; ma non già alla ta-<emph.end type="italics"/> <arrow.to.target n="note282"></arrow.to.target><lb/><emph type="italics"/>glia di &longs;opra; &longs;arà il pe&longs;o B alla po&longs;&longs;anza di C, come il cinque al due, cioè dop­<lb/>pio &longs;e&longs;quialtero.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note281"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note282"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>&<emph.end type="italics"/> 16. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Et per lo contrario ritrouaremo la proportione molteplice &longs;opraparticolare della po&longs;­<lb/>&longs;anza al pe&longs;o; & come nelle altre &longs;i mo&longs;trerà co&longs;i e&longs;&longs;ere lo &longs;patio della po&longs;&longs;anza <lb/>che moue allo &longs;patio del pe&longs;o, come il pe&longs;o alla po&longs;&longs;anza, che lo &longs;ośtiene.<emph.end type="italics"/></s> </p> <p type="main"> <s>Con l'i&longs;te&longs;&longs;o modo ritrouaremo ancora ogni proportione &longs;opra­<lb/>partiente; come &longs;e la proportione, laquale ha la po&longs;&longs;anza co'l <lb/>pe&longs;o, &longs;arà doppia &longs;oprabipartiente, come l'otto al tre. </s> </p> <p type="main"> <s><emph type="italics"/>Faccia&longs;i la po&longs;&longs;anza po&longs;ta in A &longs;o&longs;tenente il pe&longs;o B vn'ottauo del pe&longs;o B, &<emph.end type="italics"/> <arrow.to.target n="note283"></arrow.to.target><lb/><emph type="italics"/>la po&longs;&longs;anza di C &longs;ia vn terzo della po&longs;&longs;anza di A; &longs;arà il pe&longs;o B alla po&longs;&longs;an <lb/>za di C, come l'otto al tre. </s> <s>& per lo contrario ritroueremo ogni proportione mol<emph.end type="italics"/> <arrow.to.target n="note284"></arrow.to.target> <pb/><emph type="italics"/>replice &longs;oprapartiente della po&longs;&longs;anza al pe&longs;o. </s> <s>& come nelle altre ritrouaremo co&longs;i <lb/>e&longs;&longs;ere il pe&longs;o alla po&longs;&longs;anza che lo &longs;o&longs;tiene, come lo &longs;patio della po&longs;&longs;anza che moue <lb/>allo &longs;patio del pe&longs;o.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note283"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note284"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 17. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Ma egli è da notare, che benche più volte &longs;ia &longs;tato detto nelle demo&longs;irationi prece­<lb/>denti, la po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o e&longs;&longs;ere due volte tanto quanto e&longs;&longs;o pe&longs;o, ò tre, <lb/>& co&longs;i di mano in mano, come nella decimaquinta di questo è &longs;tato mo&longs;trato; non­<lb/>dimeno percioche la po&longs;&longs;anza &longs;o&longs;liene non &longs;olamente il pe&longs;o, ma la taglia ancora, <lb/>però egli pare, che &longs;ia me&longs;tieri porre la po&longs;&longs;anza di molto maggiore virtù, & di pro­<lb/>portione maggiore ver&longs;o il pe&longs;o. </s> <s>ilche è vero, &longs;<gap/> vogliamo con&longs;iderare etiandio la <lb/>grauezza della taglia. </s> <s>Ma percioche cerchiamo la proportione che è fra la po&longs;­<lb/>&longs;anza & il pe&longs;o, però habbiamo trala&longs;ciato cote&longs;ta grauezza della taglia, laquale <lb/>&longs;e alcuno vorrà anche con&longs;iderare, alla po&longs;&longs;anza potrà aggiungere &longs;orza che &longs;ia <lb/>eguale alla taglia. </s> <s>ilche mede&longs;imamente &longs;i potrà o&longs;&longs;eruare nella corda. </s> <s>& &longs;i co­<lb/>me habbiamo <gap/>ò con&longs;iderato nella decimaquinta, l'i&longs;te&longs;&longs;o parimente nelle altre po­<lb/>tremo con&longs;iderare.<emph.end type="italics"/></s> </p> <pb pagenum="98"/> <p type="main"> <s><emph type="italics"/>Egli è me&longs;tieri &longs;apere etiandio, che &longs;i come tut­<lb/>te le proportioni tra la po&longs;&longs;anza, & il pe&longs;o <lb/>&longs;ono &longs;tate ritrouate con vna &longs;ola corda: co&longs;i <lb/>ancora potranno&longs;i le i&longs;te&longs;&longs;e ritrouare con più <lb/>corde, & con più taglie. </s> <s>come &longs;e vorremo <lb/>ritrouare la proportione molteplice &longs;oprapar <lb/>ticolare con più corde, cioè &longs;e la proportio­<lb/>ne, laquale hà il pe&longs;o alla po&longs;&longs;anza che lo &longs;o <lb/>&longs;tiene &longs;arà doppia &longs;e&longs;quialtera, come cinque <lb/>à due; bi&longs;ogna comporre que&longs;ta proportione <lb/>da più proportioni come per gratia di e&longs;&longs;em­<lb/>pio dalla proportione &longs;e&longs;quiquarta, che è il <lb/>cinque al quattro, & dalla doppia, che è il <lb/>quattro al due. </s> <s>Ponga&longs;i dunque la po&longs;&longs;an-<emph.end type="italics"/> <arrow.to.target n="note285"></arrow.to.target><lb/><emph type="italics"/>za di A che &longs;o&longs;tenga il pe&longs;o B, alla qua­<lb/>le il pe&longs;o habbia la proportione di vna volta <lb/>& vn quarto, come cinque à quattro: da<emph.end type="italics"/> <arrow.to.target n="note286"></arrow.to.target><lb/><emph type="italics"/>poi con vn'altra corda &longs;i troui la po&longs;&longs;anza <lb/>di C, della quale &longs;ia doppia la po&longs;&longs;anza di <lb/>A. & percioche il B all' A è come cin­<lb/>que à quattro: & l' A. al C come il quat­<lb/>tro al due: &longs;arà la po&longs;&longs;anza di B alla po&longs;­<lb/>&longs;anza di C come il cinque al due; cioè ha­<lb/>urà la proportione doppia &longs;e&longs;quialtera.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note285"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 21. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note286"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Et è da ßotare pote &longs;i trouar' anche que&longs;ta pro­<lb/>portione, &longs;e comporremo la proportione di <lb/>cinque à due da più, come cinque à quindici, <lb/>& il quindici al venti, & il venti al due. </s> <s>Et <lb/>in que&longs;to modo ritroueremo non &longs;olo ogni al­<lb/>tra proportione, ma qualunque &longs;i &longs;ia in mol­<lb/>ti, & in&longs;initi modi ritroueremo. </s> <s>percioche <lb/>ogni proportione &longs;i può comporre di propor­<lb/>tioni in&longs;inite. </s> <s>come è manife&longs;to nel commen­<lb/>tario di Eutocio nella quarta propo&longs;itione del <lb/>&longs;econdo libro di Archimede della s&longs;era, & <lb/>Cilindro.<emph.end type="italics"/></s> </p> <p type="main"> <s>Po&longs;&longs;iamo ancora v&longs;are più corde: & <lb/>adoperare le taglie di &longs;otto &longs;ola­<lb/>mente, ouero quelle di &longs;opra. </s> </p> <pb pagenum="99"/> <p type="main"> <s><emph type="italics"/>Sia poi il pe&longs;o A legato alla fune, la­<lb/>quale &longs;ia inuolta d'intorno alla girel­<lb/>la della taglia di &longs;opra, & rilegata in <lb/>B, & &longs;ia la po&longs;&longs;anza di C che &longs;o <lb/>&longs;tenga il pe&longs;o A; &longs;arà la po&longs;&longs;anza <lb/>di C due volte tanto quanto il pe&longs;o<emph.end type="italics"/> <arrow.to.target n="note287"></arrow.to.target><lb/><emph type="italics"/>A: dapoi C &longs;ia rilegata ad vn'al­<lb/>tra fune, laquale &longs;ia rinuolta d'intor <lb/>no la girella d'vn'altra taglia, & ri­<lb/>legata in D; &longs;arà la po&longs;&longs;anza di E<emph.end type="italics"/> <arrow.to.target n="note288"></arrow.to.target><lb/><emph type="italics"/>due volte tanto quanto la po&longs;&longs;anza <lb/>di C. </s> <s>Per laqual co&longs;a la po&longs;&longs;anza <lb/>di E &longs;arà quattro volte tanto quan­<lb/>to il pe&longs;o A. </s> <s>Et &longs;e dauantaggio <lb/>lo E &longs;i rilegherà ad vn'altra fune, <lb/>laquale &longs;ia inuolta dintorno' alla gi­<lb/>rella d'vn'altra taglia ancora, & &longs;ia <lb/>rilegata in F; &longs;arà la po&longs;&longs;anza di G <lb/>due volte tanto quanto la po&longs;&longs;anza <lb/>di E. </s> <s>Adunque la po&longs;&longs;anza po&longs;ta <lb/>in G è otto volte tanto quanto il pe <lb/>&longs;o A; & co&longs;i in in&longs;inito ritrouere­<lb/>mo &longs;empre la po&longs;&longs;anza e&longs;&longs;ere due vol <lb/>te tanto quanto la po&longs;&longs;anza prece­<lb/>dente.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note287"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 15. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note288"></margin.target><emph type="italics"/>Per la isse&longs; &longs;a.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ma &longs;e in G fo&longs;&longs;e la po&longs;&longs;anza che moue, <lb/>&longs;arà lo &longs;patio del pe&longs;o otto volte tan <lb/>to quanto lo &longs;patio della po&longs;&longs;anza po <lb/>&longs;tain G: percioche lo &longs;patio del pe-<emph.end type="italics"/> <arrow.to.target n="note289"></arrow.to.target><lb/><emph type="italics"/>&longs;o A è due volte tanto quanto lo <lb/>&longs;patio della po&longs;&longs;anza posta in C, & <lb/>il C è due volte tanto quanto è lo <lb/>&longs;patio di e&longs;&longs;o E. </s> <s>Per laqual co&longs;a lo <lb/>&longs;patio del pe&longs;o A &longs;arà quattro vol <lb/>te tanto quanto lo &longs;patio della po&longs;&longs;an <lb/>za di E. &longs;imilmente percioche lo <lb/>&longs;patio di E è due volte tanto quan­<lb/>to è lo &longs;patio della po&longs;&longs;anza po&longs;ta in <lb/>G; &longs;arà dunque lo &longs;patio del pe&longs;o A <lb/>otto volte tanto quanto lo &longs;patio della <lb/>po&longs;&longs;anza po&longs;ta in G.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note289"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 16. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Da que&longs;te co&longs;e è manife&longs;to, che &longs;empre lo &longs;patio della po&longs;&longs;anza <lb/>che moue ha proportione maggiore ver&longs;o lo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o, di quel che ha il pe&longs;o ver&longs;o la mede&longs;ima po&longs;&longs;anza. </s> </p> <p type="main"> <s><emph type="italics"/>Que&longs;to è chiaro da quelle co&longs;e lequali &longs;ono &longs;tate dette nel corollario della quarta pro­<lb/>po&longs;itione di que&longs;to nella leua.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE XXVII.</s> </p> <p type="head"> <s>PROBLEMA.</s> </p> <p type="main"> <s>Che &longs;i moua vn pe&longs;o dato da vna po&longs;&longs;anza data con le taglie. </s> </p> <p type="main"> <s><emph type="italics"/>La po&longs;&longs;anza data ò che ella è maggiore, ouero eguale, ò pure minore del pe&longs;o dato.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Se è maggiore, all'hora la po&longs;­<lb/>&longs;anza, &longs;enza altro &longs;tromen­<lb/>to, ò fune inuolta d'intor­<lb/>no alla girella della taglia<emph.end type="italics"/><lb/> <arrow.to.target n="note290"></arrow.to.target> <emph type="italics"/>appiccata di &longs;opra, mouerà <lb/>il pe&longs;o dato. </s> <s>percio che po&longs; <lb/>&longs;anza minore della data pe <lb/>&longs;a tanto quanto il pe&longs;o, <lb/>adunque la data, che è mag <lb/>giore mouerà. </s> <s>L'i&longs;te&longs;&longs;o &longs;i <lb/>può fare in tutte le propo­<lb/>&longs;itioni nelle quali la po&longs;&longs;an­<lb/>za, che &longs;o&longs;tiene il pe&longs;o è &longs;ta <lb/>ta dimo&longs;trata ò eguale, ò <lb/>minore del pe&longs;o.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note290"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 1. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <pb pagenum="100"/> <p type="main"> <s><emph type="italics"/>Ma &longs;e eguale mouerà il <lb/>pe&longs;o e&longs;&longs;endo la fune <lb/>inuolta d'intorno al <lb/>la girella della ta­<lb/>glia legata al pe&longs;o,<emph.end type="italics"/> <arrow.to.target n="note291"></arrow.to.target><lb/><emph type="italics"/>percio che la po&longs;&longs;an­<lb/>za che &longs;o&longs;tiene il pe <lb/>&longs;o è la metà del pe­<lb/>&longs;o. </s> <s>la po&longs;&longs;anza dun­<lb/>que eguale al. </s> <s>pe&longs;o <lb/>mouerà il pe&longs;o da­<lb/>to. </s> <s>ilche parimen­<lb/>te &longs;i puote fare &longs;e­<lb/>condo le propo&longs;itio­<lb/>ni, nellequali &longs;i è <lb/>mo&longs;trato la po&longs;&longs;an­<lb/>za e&longs;&longs;ere minore del <lb/>pe&longs;o.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note291"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e è minore, &longs;ia il pe&longs;o <lb/>dato come &longs;e&longs;&longs;anta, & la <lb/>po&longs;&longs;anza che moue &longs;ia da­<lb/>ta come tredici. </s> <s>Troui&longs;i <lb/>la po&longs;&longs;anza di A, che &longs;o <lb/>&longs;tenga il pe&longs;o B, laquale<emph.end type="italics"/><lb/> <arrow.to.target n="note292"></arrow.to.target> <emph type="italics"/>&longs;ia vn quinto del pe&longs;o </s> <s>/>& percioche la po&longs;&longs;anza <lb/>di A che &longs;o&longs;tiene il pe&longs;o <lb/>è come dodici; adunque <lb/>po&longs;&longs;anza maggiore di do­<lb/>dici po&longs;ta in A mouerà <lb/>il pe&longs;o B. </s> <s>Per laqual c<gap/><lb/>&longs;ala po&longs;&longs;anza come tredi­<lb/>ci po&longs;ta in A mouerà il <lb/>pe&longs;o B. che bi&longs;ognaua <lb/>fare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note292"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Egli è parimente da auerti­<lb/>re nel mouere i pe&longs;i, che <lb/>la po&longs;&longs;anza alcuna volta <lb/>meglio for&longs;e moue mouen­<lb/>do&longs;i in giù, che mouendo&longs;i <lb/>in sù. </s> <s>come volga&longs;i dauan <lb/>taggio la fune d'intorno ad <lb/>vn'altra girella della ta­<lb/>glia di &longs;opra, il cui centro <lb/>&longs;ia C, & la fune per­<lb/>uenga in D; &longs;arà la po&longs;-<emph.end type="italics"/><lb/> <arrow.to.target n="note293"></arrow.to.target> <emph type="italics"/>&longs;anza di D &longs;o&longs;tenente il <lb/>pe&longs;o B &longs;imilmente dodi­<lb/>ci, &longs;i come ella erain A. </s> <s><lb/>Però la po&longs;&longs;anza di tredici <lb/>po&longs;ta in D mouerà il pe­<lb/>&longs;o B. </s> <s>& percioche &longs;i <lb/>moue in giù, for&longs;e tirerà <lb/>più facilmente, che &longs;e fo&longs;­<lb/>&longs;e po&longs;ta in A, ma il tem <lb/>po è l'i&longs;te&longs;&longs;o, &longs;i come egli <lb/>era etiandio in A.<emph.end type="italics"/></s> </p> <pb pagenum="101"/> <p type="margin"> <s><margin.target id="note293"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 5. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="head"> <s>PROPOSITIONE XXVIII.</s> </p> <p type="head"> <s>PROBLEMA.</s> </p> <p type="main"> <s>Sia propo&longs;to à noi il fare, che la po&longs;&longs;anza mouente il pe&longs;o, & il <lb/>pe&longs;o &longs;i mouano per gli <lb/>&longs;patij dati, i quali &longs;ia­<lb/>no fra loro commen­<lb/>&longs;urabili. </s> </p> <p type="main"> <s><emph type="italics"/>Sia dato lo &longs;patio della po&longs;&longs;anza<emph.end type="italics"/> <arrow.to.target n="note294"></arrow.to.target><lb/><emph type="italics"/>come tre, & del pe&longs;o come <lb/>quattro. </s> <s>ritroui&longs;i la po&longs;&longs;anza <lb/>di A &longs;o&longs;tenente il pe&longs;o B, la <lb/>quale &longs;ia vna volta, & vn ter­<lb/>zo quanto il pe&longs;o, come quat­<lb/>tro à tre. </s> <s>Se dunque in A<emph.end type="italics"/> <arrow.to.target n="note295"></arrow.to.target><lb/><emph type="italics"/>fo&longs;&longs;e la po&longs;&longs;anza mouente il <lb/>pe&longs;o; &longs;arebbe lo &longs;patio del pe­<lb/>&longs;o vna volta, & vn terzo <lb/>quanto lo &longs;patio della po&longs;&longs;an­<lb/>za, cioè come quattro à tre; <lb/>che bi&longs;ognaua fare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note294"></margin.target><emph type="italics"/>Per la.<emph.end type="italics"/> 22. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note295"></margin.target><emph type="italics"/>Per <gap/>istof-fa.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Ciò po&longs;&longs;iamo menar ad effetto con <lb/>vna &longs;ola fune per le co&longs;e det­<lb/>te nella vige&longs;ima &longs;econda, & <lb/>nella vige&longs;ima quinta di que­<lb/>&longs;to. </s> <s>che &longs;e ciò vorremo fare <lb/>con più funi, potremo porlo <lb/>in opra non &longs;olo con molti, <lb/>ma con modi infiniti, come di <lb/>&longs;opra è detto. </s> <s>Per laqual co­<lb/>&longs;a ciò ben po&longs;siamo affermare,<emph.end type="italics"/> <arrow.to.target n="note296"></arrow.to.target><lb/><emph type="italics"/>che pare co&longs;a marauiglio&longs;a, <lb/>cioè.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note296"></margin.target><emph type="italics"/>Nella<emph.end type="italics"/> 26. <emph type="italics"/>di questo.<emph.end type="italics"/></s> </p> <p type="head"> <s>CROLLARIO I.</s> </p> <p type="main"> <s>Da que&longs;te co&longs;e e&longs;&longs;ere manife&longs;to, Qualunque data proportione <lb/>ne i numeri tra il pe&longs;o, & la po&longs;&longs;anza; & tra lo &longs;patio del pe&longs;o <lb/>mo&longs;&longs;o, & lo &longs;patio della po&longs;&longs;anza mo&longs;&longs;a; poter&longs;i trouare con <lb/>le taglie in modi infiniti. </s> </p> <p type="head"> <s>COROLLARIO II.</s> </p> <p type="main"> <s>Dalle co&longs;e dette è manife&longs;to etiandio che quanto più facilmen­<lb/>te &longs;i moue il pe&longs;o, tanto maggiore e&longs;&longs;ere etiandio il tempo; <lb/>ma quanto più difficilmente, tanto minore e&longs;&longs;ere: & co&longs;i per <lb/>lo contrario. </s> </p> <p type="head"> <s>IL FINE DELLA TAGLIA.</s> </p> <pb pagenum="102"/> <p type="head"> <s>DELL'ASSE</s> </p> <p type="head"> <s>NELLA ROTA.</s> </p> <figure></figure> <p type="main"> <s>La fabrica, & compo&longs;itione di que&longs;to i&longs;trumento <lb/>in&longs;egna Pappo nell'ottauo libro delle raccolte ma­<lb/>tematiche: & chiama a&longs;&longs;e AB, & timpano CD <lb/>d'intorno al centro mede&longs;imo (che noi diremo ro­<lb/>ta) & noma&longs;citale quei ba&longs;toni i quali &longs;ono fic­<lb/>cati ne'buchi della rota notate per EFGH, & le altre &longs;uc­<lb/>ce&longs;&longs;iuamente, che noi pur diremo raggi. </s> <s>talche la po&longs;&longs;anza, <pb/>laquale è &longs;empre ne i raggi, come in F, mentre ella volge <lb/>intorno la rota, & l'a&longs;&longs;e, moua anco in sù il pe&longs;o K appicca­<lb/>to all'a&longs;&longs;e con la corda LM riuolta d'intorno all'a&longs;&longs;e. </s> <s>A noi <lb/>re&longs;ta dunque, di mo&longs;trare, perche i gran pe&longs;i da piccola forza, <lb/> <arrow.to.target n="fig106"></arrow.to.target><lb/>& in che modo etiandio &longs;i mouano con que&longs;to i&longs;trumento: <lb/>& di più manife&longs;tare la ragione del tempo, & dello &longs;patio del­<lb/>la po&longs;&longs;anza mouente, & del pe&longs;o mo&longs;&longs;o fra loro, & ridurre l'v­<lb/>&longs;o di cote&longs;to i&longs;trumento alla leua. </s> </p> <pb pagenum="103"/> <figure id="fig106"></figure> <p type="head"> <s>PROPOSITIONE I.</s> </p> <p type="main"> <s>La po&longs;&longs;anza &longs;o&longs;tenente il pe&longs;o con l'a&longs;&longs;e nella rota, ha la propor­<lb/>tione mede&longs;ima al pe&longs;o, che il mezo diametro dell'a&longs;&longs;e al me­<lb/>zo diametro della rota in&longs;ieme co'l raggio. </s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Sia il diametro dell'a&longs;&longs;e AB, & il &longs;uo centro C; &longs;ia il diametro della rota DCE <lb/>d'intorno al centro mede&longs;imo; & &longs;iano AB DE nell i&longs;te&longs;&longs;a linea retta; &longs;iano <lb/>dopo li raggi eguali tra loro, & egualmente di&longs;tanti DF GH, & gli altri ne' bu­<lb/>chi della rota; & &longs;ia FE egualmente di&longs;tante dall'orizonte, & il pe&longs;o K &longs;ia <pb/>appiccato alla corda BL volubile d'intorno all'a&longs;&longs;e. </s> <s>& la po&longs;&longs;anza po&longs;ta in F <lb/>&longs;o&longs;tenga il pe&longs;o K. </s> <s>Dico che la po&longs;&longs;anza in F co&longs;i &longs;i hà al pe&longs;o K, come CB <lb/>à CF. </s> <s>Faccia&longs;i come CF à CB, co&longs;i il pe&longs;o K ad vn altro pe&longs;o come M, il <lb/>quale &longs;ia appi<gap/>ato in F. </s> <s>& percioche i pe&longs;i MK &longs;ono appiccati in -FB; &longs;arà <lb/>FB come leua, ouero bilancia; ma percioche il C è punto immobile, d'intorno<emph.end type="italics"/><lb/> <arrow.to.target n="fig107"></arrow.to.target><lb/> <arrow.to.target n="note297"></arrow.to.target> <emph type="italics"/>alquale l'a&longs;&longs;e, & la rota &longs;i riuolgono; &longs;arà C il &longs;o&longs;tegno della leua FB, ouero il <lb/>centro della bilancia. </s> <s>& per e&longs;&longs;ere co&longs;i CF à CB come K ad M, i pe&longs;i KM <lb/>pe&longs;eranno egualmente. </s> <s>La po&longs;&longs;anza dunque di F &longs;o&longs;tenente il pe&longs;o K contra­<lb/>pe&longs;erà egualmente con e&longs;&longs;o pe&longs;o K accioche egli non chini al ba&longs;&longs;o, & &longs;arà eguale <lb/>ad M. </s> <s>Percioche la po&longs;&longs;anza opera il mede&longs;imo che il pe&longs;o M. dunque il pe&longs;o K<emph.end type="italics"/> <pb pagenum="104"/><emph type="italics"/>&longs;arà alla po&longs;&longs;anza di F, come CF à CB, & conuertendo la po&longs;&longs;anza &longs;arà al <lb/>pe&longs;o, come CB à CF, cioè il mezo diametro dell'a&longs;&longs;e al mezo diametro della rota<emph.end type="italics"/> <arrow.to.target n="note298"></arrow.to.target><lb/><emph type="italics"/>in&longs;ieme co'l raggio DF. &longs;imilmente mo&longs;trera&longs;&longs;i anco, che &longs;e la po&longs;&longs;anza &longs;o&longs;tenente <lb/>il pe&longs;o fo&longs;&longs;e in Q, all'hora &longs;o&longs;terrebbe con la leua <expan abbr="Cq;">Cque</expan> & haurebbe quella pro-<emph.end type="italics"/> <arrow.to.target n="note299"></arrow.to.target><lb/><emph type="italics"/>portione al pe&longs;o, che CB haue à <expan abbr="Cq;">Cque</expan> cioè il mezo diametro dell'a&longs;&longs;e al mezo dia­<lb/>metro della rota in&longs;ieme co'l raggio EQ, che bi&longs;ognaua dimo&longs;trare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note297"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del<emph.end type="italics"/> 1. <emph type="italics"/>d' Ar chimede del le co&longs;e che pe&longs;anoegual mense.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note298"></margin.target><emph type="italics"/>Per lo corol lario della<emph.end type="italics"/> 4. <emph type="italics"/>del<emph.end type="italics"/> 5.</s> </p> <p type="margin"> <s><margin.target id="note299"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>di questo della leua.<emph.end type="italics"/></s> </p> <figure id="fig107"></figure> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Egli è manife&longs;to che la po&longs;&longs;anza &longs;empre è minore del pe&longs;o. </s> </p> <p type="main"> <s><emph type="italics"/>Percio che il mezo diametro dell'a&longs;&longs;e &longs;empre è minore del mezo diametro della rota. <lb/>& la po&longs;&longs;anza in tanto è minore del pe&longs;o, in quanto il mezo diametro dell'a&longs;&longs;e è mi <lb/>nore del mezo diametro della rota in&longs;ieme co'l raggio. </s> <s>Per laqual co&longs;a quanto è più <lb/>lungo CF, ouero <expan abbr="Cq;">Cque</expan> & quanto è più corto CB, tanto anco &longs;empre minore po&longs;&longs;an <lb/>za po&longs;ta in F, ouero in Q, &longs;o&longs;tenterà il pe&longs;o K. percioche quanto minore è CB, <lb/>tanto il mezo diametro dell'a&longs;&longs;e, haurà proportione minore al mezo diametro della <lb/>rota in&longs;ieme co'l raggio.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>In que&longs;to loco occorre da e&longs;&longs;ere con&longs;iderato, che &longs;e il pe&longs;o &longs;arà appiccato in vn'altro <lb/>raggio, come in T, che &longs;o&longs;tenga il pe&longs;o K, in modo cioè, che il pe&longs;o appiccato in <lb/>T, & il pe&longs;o K po&longs;to d'intorno all'a&longs;&longs;e rimangano; &longs;arà il pe&longs;o in T più graue del <lb/>pe&longs;o M appiccato in F. </s> <s>Percioche &longs;ia congiunta TB, & dal punto C &longs;ia <lb/>tirata la CI à piombo dell'orizonte, laquale tagli la TB in I; & alla fine con <lb/>giunga&longs;i TC, laquale &longs;arà eguale à CF. </s> <s>Et percioche i pe&longs;i &longs;ono appiccati in <lb/>TB &longs;i haueranno in modo come &longs;e haue&longs;&longs;ero i centri delle grauezze loro in TB, <lb/>come dianzi fu detto. </s> <s>& percbe rimangono, &longs;arà il punto I per la prima di que­<lb/>&longs;to della bilancia, il centro della grauezza di ambidue in&longs;ieme, per e&longs;&longs;ere CI à piom<emph.end type="italics"/> <arrow.to.target n="note300"></arrow.to.target><lb/><emph type="italics"/>bo dell'orizonte. </s> <s>Ma percioche l'angolo BCI è retto, &longs;ara BIC acuto, & la <lb/>linea BI &longs;arà maggiore di e&longs;&longs;a BC. </s> <s>Per laqual co&longs;a l'angolo CIT &longs;arà ottu&longs;o,<emph.end type="italics"/> <arrow.to.target n="note301"></arrow.to.target><lb/><emph type="italics"/>& perciò la linea CT &longs;arà maggiore di TI. </s> <s>Et concio&longs;ia che CT &longs;ia maggiore <lb/>di TI, & IB maggiore di BC; haurà TC proportione maggiore à CB, <lb/>che TI ad IB; & conuertendo BC haurà proportione minore à CT, cioè <lb/>à CF, che BI ad IT, come per la vige&longs;ima&longs;e&longs;ta del quinto de gli elementi; <lb/>(&longs;econdo il Commandino) è mani&longs;e&longs;to. </s> <s>Ma percioche il punto I è centro della <lb/>grauezza de' pe&longs;i &longs;tanti in TB, &longs;arà il pe&longs;o po&longs;to in T al pe&longs;o po&longs;to in B, come<emph.end type="italics"/> <arrow.to.target n="note302"></arrow.to.target><lb/><emph type="italics"/>BI ad IT. ma il pe&longs;o in F &longs;i hà al pe&longs;o mede&longs;imo in B, come BC à CF; <lb/>dunque il pe&longs;o in T haurà proportione maggiore al pe&longs;o in B, che il pe&longs;o in F <lb/>a'li&longs;te&longs;&longs;o pe&longs;o in B. </s> <s>adunque &longs;arà più graue il pe&longs;o in T, che il pe&longs;o in F.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note300"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 29. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note301"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 13. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note302"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 6. <emph type="italics"/>del<emph.end type="italics"/> 1. <emph type="italics"/>di Ar chimede del le co&longs;e che pe&longs;ano egual mente.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Che &longs;e in loco del pe&longs;o in T &longs;i porrà vna po&longs;&longs;anza animata, che &longs;o&longs;tenga il pe&longs;o K, <lb/>laquale in maniera &longs;i inchini, come &longs;e vole&longs;&longs;e andare al centro del mondo, come di<emph.end type="italics"/> <arrow.to.target n="note303"></arrow.to.target><lb/><emph type="italics"/>&longs;ua propria natura &longs;à il pe&longs;o appiccato in T; &longs;arà que&longs;ta &longs;te&longs;&longs;a eguale al pe&longs;o ap- <pb/>piccato in T, altramente non &longs;o&longs;tentarebbe, laquale veramente &longs;arà maggiore <lb/>della po&longs;&longs;anza collocata in F. </s> <s>percioche &longs;i come &longs;i ha il pe&longs;o di T al pe&longs;o di F, <lb/>co&longs;i ha&longs;&longs;i anco la po&longs;&longs;anza di T alla po&longs;&longs;anza di F, per e&longs;&longs;ere le po&longs;&longs;anze eguali <lb/>a' pe&longs;i. </s> <s>Ma &longs;e cia&longs;cheduna po&longs;&longs;anza pre&longs;a &longs;eparatamente &longs;o&longs;tenente il pe&longs;o tanto <lb/>in T quanto in F, &longs;econdo la circonferenza THFN, &longs;i vole&longs;&longs;e mouere, come <lb/>&longs;e il raggio fo&longs;&longs;e pre&longs;o con vna mano; all'hora la mede&longs;ima po&longs;&longs;anza po&longs;ta in F, <lb/>ouero in T, potrà &longs;o&longs;tenere l'i&longs;te&longs;&longs;o pe&longs;o K; concio&longs;ia, che ponga&longs;i pure nella &longs;tre <lb/>mità di qual &longs;i voglia raggio, &longs;empre verrà ad e&longs;&longs;ere egualmente di&longs;tante dall'i&longs;te&longs;­<lb/>&longs;o centro C, & ad hauere la &longs;ua inclinatione &longs;econdo la circon&longs;erenza i&longs;te&longs;&longs;a egual <lb/>mente di&longs;tante &longs;empre dal centro mede&longs;imo. </s> <s>ne come fa il pe&longs;o di &longs;ua propria na­<lb/>tura più de&longs;idera e&longs;&longs;ere portata nel centro, che mouer&longs;i in cerchio: percioche riguar­<lb/>da l'vno, & l'altro, ouero qual &longs;i voglia altro mouimento &longs;enza veruna differenza <lb/>in tutto. </s> <s>Per laqual co&longs;a non i&longs;ta il fatto nel modo i&longs;te&longs;&longs;o, &longs;e ouero i pe&longs;i, ouero le <lb/>po&longs;&longs;anze animate &longs;aranno po&longs;te ne' luoghi mede&longs;imi per far l'i&longs;te&longs;&longs;o officio.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note303"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 10. <emph type="italics"/>del<emph.end type="italics"/> 5.</s> </p> <p type="main"> <s><emph type="italics"/>Mala po&longs;&longs;anza moue il pe&longs;o con la leua FB, cioè mentre la po&longs;&longs;anza di F volge in­<lb/>torno la rota, gira intorno anche l'a&longs;&longs;e, & FB &longs;i fà come leua, il cui &longs;o&longs;tegno è C; <lb/>la po&longs;&longs;anza mouente in F, & il pe&longs;o è appiccato in B: & mentre il punto F <lb/>peruiene in N il punto H &longs;arà in F, & il punto B &longs;arà in O; per modo che <lb/>la tirata linea NO pa&longs;&longs;i per C; & nell'i&longs;te&longs;&longs;o tempo il pe&longs;o K &longs;arà mo&longs;&longs;o in P, <lb/>per modo che OBP &longs;ia eguale ad e&longs;&longs;o BL, e&longs;&longs;endo la i&longs;te&longs;&longs;a corda.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Dapoi dalla quarta di que&longs;to della leua ageuolmente caueremo co&longs;i e&longs;&longs;ere lo &longs;patio del <lb/>la po&longs;&longs;anza che moue allo &longs;patio del pe&longs;o mo&longs;&longs;o, come il mezo diametro della rota <lb/>in&longs;ieme co'l raggio al mezo diametro dell'a&longs;&longs;e, cioè come CF à CB; per e&longs;&longs;ere<emph.end type="italics"/><lb/> <arrow.to.target n="note304"></arrow.to.target> <emph type="italics"/>la circonferenza FN ver&longs;o BO, come CF à CB. </s> <s>Et percioche BL è egua <lb/>le ad OBP, leuata via la commune BP, &longs;arà OB eguale ad e&longs;&longs;a PL. </s> <s>Per la qual <lb/>co&longs;a FN che è lo &longs;patio della po&longs;&longs;anza ver&longs;o PL &longs;patio del pe&longs;o, &longs;arà come CF <lb/>à CB, cioè il mezo diametro della rota in&longs;ieme co'l raggio al mezo diametro del­<lb/><gap/>a&longs;&longs;e. </s> <s>Laqual co&longs;a parimente mo&longs;trera&longs;&longs;i, &longs;tando la po&longs;&longs;anza in Q, ouero in qual &longs;i <lb/>voglia altro raggio, come in S. </s> <s>concio&longs;ia, che e&longs;&longs;endo li raggi fra lcro eguali, & <lb/>egualmente di&longs;tanti; &longs;ia doue &longs;i voglia la po&longs;&longs;anza mo&longs;&longs;a con velocità eguale, tra­<lb/>pa&longs;&longs;erà &longs;empre in tempo eguale &longs;patio eguale, cioè da Q in R, ouero da S in T <lb/>&longs;i mouerà nel mede&longs;imo tempo, che da F in N. ma in quel tempo che la po&longs;&longs;anza <lb/>&longs;i moue da F in N, nel mede&longs;imo in tutto anco il pe&longs;o K da L &longs;i moue in </s> <s>/>adunque &longs;ia doue &longs;i voglia la po&longs;&longs;anza, &longs;arà lo &longs;patio della po&longs;&longs;anza allo &longs;patio del <lb/>pe&longs;o mo&longs;&longs;o, come CF à CB, cioè come il mezo diametro della rota co'l raggio al <lb/>mezo diametro dell'a&longs;&longs;e.<emph.end type="italics"/></s> </p> <pb pagenum="105"/> <p type="margin"> <s><margin.target id="note304"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 4. <emph type="italics"/>di questo della leua.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO I.</s> </p> <p type="main"> <s>Da que&longs;te co&longs;e è manife&longs;to, che co&longs;i è il pe&longs;o alla po&longs;&longs;anza &longs;o&longs;te­<lb/>nente il pe&longs;o, come lo &longs;patio della po&longs;&longs;anza mouente allo &longs;pa <lb/>tio del pe&longs;o mo&longs;&longs;o. </s> </p> <figure></figure> <p type="head"> <s>COROLLARIO II.</s> </p> <p type="main"> <s>Egli è manife&longs;to etiandio, che lo &longs;patio della po&longs;&longs;anza mouen­<lb/>te hà &longs;empre maggiore proportione allo &longs;patio del pe&longs;o mo&longs;­<lb/>&longs;o, che il pe&longs;o alla &longs;te&longs;&longs;a po&longs;&longs;anza. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Oltre à ciò quanto il cerchio FHN d'intorno à i raggi è più grande, tanto anco &longs;i <lb/>con&longs;umerà più tempo in mouere il pe&longs;o, pur che la po&longs;&longs;anza &longs;i moua con eguale ve-<emph.end type="italics"/><lb/> <arrow.to.target n="note305"></arrow.to.target> <emph type="italics"/>locità; & il tempo tanto &longs;arà maggiore quanto il diametro dell'vno &longs;arà maggiore <lb/>del diametro dell'altro; percioche le circonferenze de'cerchi &longs;i hanno come i diame <lb/>tri. </s> <s>& concio&longs;ia, che per la trige&longs;ima &longs;e&longs;ta del quarto libro di Pappo delle raccolte<emph.end type="italics"/><lb/> <arrow.to.target n="fig108"></arrow.to.target><lb/><emph type="italics"/>matematiche po&longs;&longs;iamo ritrouare le circonferenze eguali di due cerchi di&longs;uguali; per­<lb/>ciò ritroueremo anche il tempo à que&longs;to modo delle portioni di&longs;uguali de' cerchi. </s> <s>Ma <lb/>per lo contrario quanto &longs;arà maggiore la circonferenza dell'a&longs;&longs;e, il pe&longs;o mouera&longs;&longs;i <lb/>più pre&longs;to in sù, percioche maggior parte della corda BL in vno giro compiuto, &longs;i <lb/>riuolge d'intorno al cerchio ABO, che &longs;e fo&longs;&longs;e minore, per e&longs;&longs;ere la corda inuolta <lb/>eguale alla circonferenza del cerchio, d'intorno alquale &longs;i riuolge.<emph.end type="italics"/></s> </p> <pb pagenum="106"/> <p type="margin"> <s><margin.target id="note305"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 23. <emph type="italics"/>dell'estano li bre di Pappe.<emph.end type="italics"/></s> </p> <figure id="fig108"></figure> <p type="head"> <s>COROLLARIO.</s> </p> <p type="main"> <s>Da que&longs;te co&longs;e è manife&longs;to, che quanto più ageuolmente &longs;i mo <lb/>ue il pe&longs;o, tanto il tempo è anco maggiore; & quanto più ma­<lb/>lageuolmente, tanto il tempo e&longs;&longs;ere minore. </s> <s>& co&longs;i per lo <lb/>contrario. </s> </p> <p type="head"> <s>PROPOSITIONE II.</s> </p> <p type="head"> <s>PROBLEMA.</s> </p> <p type="main"> <s>Far che &longs;i moua vn dato pe&longs;o, con l'a&longs;&longs;e nella rota da vna data <lb/>po&longs;&longs;anza. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il dato pe&longs;o &longs;e&longs;&longs;anta, & la po&longs;&longs;anza come dieci. </s> <s>Faccia&longs;i vna linea retta AB, laquale <lb/>&longs;i diuida in C, &longs;i fattamente che AC habbia la proportione i&longs;te&longs;&longs;a à CB, che ha <lb/>&longs;e&longs;&longs;anta à diece. </s> <s>& &longs;e CB <lb/>fo&longs;&longs;e il mezo diametro del­<lb/>l'a&longs;&longs;e, & CA il mezo dia<emph.end type="italics"/> <arrow.to.target n="note306"></arrow.to.target><lb/><emph type="italics"/>metro della rota co'raggi; <lb/>egli è chiaro, che la po&longs;&longs;an­<lb/>za come dieci po&longs;ta in A<emph.end type="italics"/> <arrow.to.target n="note307"></arrow.to.target><lb/><emph type="italics"/>pe&longs;erebbe egualmente co'l<emph.end type="italics"/><lb/> <arrow.to.target n="fig109"></arrow.to.target><lb/><emph type="italics"/>pe&longs;o &longs;e&longs;&longs;anta po&longs;to in B. </s> <s>ma pigli&longs;i tra BC qual &longs;i voglia punto, & &longs;ia D; & <lb/>faccia&longs;i BD il mezo diametro dell'a&longs;&longs;e, & DA il mezo diametro della rota co'rag <lb/>gi, & ponga&longs;i il pe&longs;o &longs;e&longs;&longs;anta in B con vna corda inuolta d'intorno all'a&longs;&longs;e, & la po&longs;<emph.end type="italics"/> <arrow.to.target n="note308"></arrow.to.target><lb/><emph type="italics"/>&longs;anza in A. </s> <s>Hor percioche AD ha proportione maggiore à DB, che AC à <lb/>CB: haur à proportione maggiore AD à DB, che il pe&longs;o &longs;e&longs;&longs;anta appiccato in <lb/>B alla po&longs;&longs;anza di dieci posta in A. </s> <s>Per laqual co&longs;a la po&longs;&longs;anza di A mouerà il <lb/>pe&longs;o di &longs;e&longs;&longs;anta con l'a&longs;&longs;e nella rota, il mezo diametro delquale è BD, & DA è <lb/>il mezo diametro della rota co'raggi. </s> <s>ilche era da far&longs;i.<emph.end type="italics"/></s> </p> <pb/> <p type="margin"> <s><margin.target id="note306"></margin.target><emph type="italics"/>Per la prece dente.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note307"></margin.target><emph type="italics"/>Per il lemma nella pri ma di questo della leua.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note308"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 11. <emph type="italics"/>di questo della leua.,<emph.end type="italics"/></s> </p> <figure id="fig109"></figure> <p type="head"> <s>Altramente.</s> </p> <p type="main"> <s>Ma Mecanicamente meglio &longs;arà in que&longs;to modo. </s> </p> <p type="main"> <s><emph type="italics"/>Ponga&longs;i l'a&longs;&longs;e, il cui mezo diametro &longs;ia BD, & il centro &longs;uo C, ilquale a&longs;&longs;e &longs;ta­<lb/>tuiremo maggiore, ò minore, come la grandezza, & grauezza del pe&longs;o ricerca. </s> <s><lb/>Allunghi&longs;i po&longs;cia la li­<lb/>nea BD fin ad A; & <lb/>faccia&longs;i BC à CA, co <lb/>me diece à &longs;e&longs;&longs;anta., & <lb/>&longs;e CA fo&longs;&longs;e il mezo dia <lb/>metro della rota co'rag <lb/>gi, la po&longs;&longs;anza di diece<emph.end type="italics"/><lb/> <arrow.to.target n="fig110"></arrow.to.target><lb/><emph type="italics"/>po&longs;ta in A pe&longs;erebbe egualmente co'l pe&longs;o di &longs;e&longs;&longs;anta po&longs;to in B. </s> <s>Ma allunghi&longs;i, <lb/>BA dalla parte di A, & in que&longs;ta allungata linea prenda&longs;i qual &longs;i voglia punto <lb/>come E, & faccia&longs;i CE il mezo diametro della rota co'raggi; & pong a&longs;i la po&longs;­<lb/>&longs;anza di diece in E; haurà EC a CB proportione maggiore, che il pe&longs;o &longs;e&longs;&longs;anta <lb/>po&longs;to in B alla po&longs;&longs;anza di diece po&longs;ta in E. </s> <s>Dunque la po&longs;&longs;anza di diece po&longs;ta in <lb/>E mouerà il pe&longs;o &longs;e&longs;&longs;anta appiccato in B, con la corda inuolta d'intorno all'a&longs;&longs;e, il <lb/>cui mezo diametro è CB, & CE è il mezo diametro della rota co i raggi. </s> <s>che bi­<lb/>&longs;ognaua fare.<emph.end type="italics"/></s> </p> <figure id="fig110"></figure> <p type="main"> <s>Sotto que&longs;ta &longs;orte d'i&longs;trumento &longs;ono gli argani, i molinelli, le tri <lb/>uelle, i timpani, ò rote co' &longs;uoi a&longs;&longs;i, ò &longs;iano dentate, ò nò, & <lb/>&longs;imili. </s> </p> <p type="main"> <s><emph type="italics"/>Ma la triuella tiene anco non &longs;o che della vite; pero che mentre moue il pe&longs;o, cioè men­<lb/>tre fora, per &longs;ua qua&longs;i natura &longs;empre trapa&longs;&longs;a viè più oltre: percioche ha qua&longs;i le <lb/>helici de&longs;critte come d'intorno ad vn cono. </s> <s>ma perche ella ha la cima acuta, &longs;i puo­<lb/>te anche ridurre commodamente alla ragione del cuneo.<emph.end type="italics"/></s> </p> <pb pagenum="107"/> <figure></figure> <p type="main"> <s>L'Autore hà qui me&longs;&longs;o que&longs;te cinque figure, lequali rappre&longs;entano cinque i&longs;trumen <lb/>ti da mouer pe&longs;i, iquali &longs;i riducono &longs;otto que&longs;ta facultà, accio che &longs;i vegga es&longs;i e&longs;­<lb/>&longs;er vna co&longs;a mede&longs;ima con l'i&longs;trumento dell'a&longs;&longs;e nella rota già dichiarato; & vi <lb/>hà po&longs;to le lettere ABC con le &longs;ue linee, per dar ad intendere, che il pe&longs;o hà la <lb/>proportione mede&longs;ima alla po&longs;&longs;anza, che lo &longs;o&longs;tiene, che hà AC à CB, & &longs;e <lb/>&longs;arà mo&longs;&longs;o il pe&longs;o da vna po&longs;&longs;anza mouente, lo &longs;patio della po&longs;&longs;anza &longs;arà &longs;imil­<lb/>mente allo &longs;patio del pe&longs;o, come AC à CB; laqual po&longs;&longs;anza deue&longs;i intendere <lb/>po&longs;ta in cima de i manichi delle &longs;tanghette di&longs;co&longs;to dal centro tanto quanto è <lb/>CA. </s> <s>Il pe&longs;o has&longs;i poi da intendere legato ad vna corda, che &longs;ia auolta d'intorno <lb/>all'a&longs;&longs;e, ilquale &longs;arà lontano dal centro tanto quanto è CB: & co&longs;i per le co&longs;e <lb/>dette in que&longs;to Trattato, la po&longs;&longs;anza che &longs;o&longs;tien haurà quella proportione al pe&longs;o, <lb/><gap/>e ha CB à CA. </s> <s>Con &longs;imile modo s'ha da intendere la figura, che hà il timpano, <lb/>con&longs;iderando che &longs;e la forza fo&longs;&longs;e nella &longs;tremità del timpano, & il pe&longs;o &longs;arebbe <lb/>auolto d'intorno all'a&longs;&longs;e. </s> <s>Quanto alla triuella, ò &longs;ucchiello che &longs;i nomi, per e&longs;­<lb/>&longs;ore vn'i&longs;trumento fatto non per &longs;o&longs;tenere, ma per mouere, egli è bi&longs;ogno, che <lb/>la po&longs;&longs;anza habbia proportione maggiore al pe&longs;o di quel che ha CB à CA per <lb/>la vndecima propo&longs;itione di que&longs;to nella leua. </s> </p> <p type="head"> <s>IL FINE DE LL'ASSE NELLA ROTA.</s> </p> <p type="head"> <s>DEL CVNEO.</s> </p> <figure></figure> <p type="main"> <s>Aristotele nelle que&longs;tioni mecaniche <lb/>nella que&longs;tione 17. afferma, che il cuneo nel <lb/>fendere vn pe&longs;o fa l'officio totalmente di due le <lb/>ue contrarie l'vna all'altra fra loro in que&longs;to <lb/>modo. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il cuneo ABC, & la &longs;ua cima B, & &longs;ia AB eguale à BC, & quel che s'ha <lb/>da fendere &longs;ia DE <lb/>FG; & &longs;ia la par <lb/>te del cuneo HBK <lb/>fra DE FG, & <lb/>HB &longs;ia eguale ad <lb/>e&longs;&longs;a BK. Percuo­<lb/>ta&longs;i, come &longs;uol far&longs;i, <lb/>il cuneo in AC, <lb/>mentre il cuneo viè <lb/>perco&longs;&longs;o in AC, &longs;i <lb/>fà AB leua, il cui <lb/>&longs;o&longs;tegno è in H, & <lb/>il pe&longs;o in B. & nel <lb/>modo i&longs;te&longs;&longs;o CB &longs;i <lb/>fa leua, il cui &longs;o&longs;te­<lb/>gno è K, & il pe­<lb/>&longs;o &longs;imilmente in B. </s> <s><lb/>Ma mentre il cu­<lb/>neo è perco&longs;&longs;o, egli <lb/>entra in e&longs;&longs;o DE<emph.end type="italics"/><lb/> <arrow.to.target n="fig111"></arrow.to.target><lb/><emph type="italics"/>FG anco con portione di &longs;e maggiore di quel che fo&longs;&longs;e prima: & &longs;ia questa por­<lb/>tione MBL; & &longs;ia MB eguale ad e&longs;&longs;a BL. & per e&longs;&longs;ere MB, & BL mag­<lb/>giori di HB BK, &longs;arà anco ML maggiore di HK. </s> <s>Mentre dunque ML &longs;arà <lb/>nel &longs;ito di HK; egli è me&longs;tieri che la fe&longs;&longs;a &longs;i faccia maggiore; & che D &longs;i moua<emph.end type="italics"/> <pb pagenum="108"/><emph type="italics"/>ver&longs;o O, & G ver&longs;o N; & quanto maggior parte del cuneo entra fra DEFG, <lb/>tanto maggior fe&longs;&longs;a &longs;i &longs;accia; & DG &longs;empre più &longs;aranno cacciati ver&longs;o ON. <lb/>dunque la parte KG che &longs;i fende mouera&longs;&longs;i dalla leua AB, ilcui &longs;o&longs;tegno è in H, <lb/>& il pe&longs;o in B; &longs;iche il punto B di e&longs;&longs;a leua AB cacci la parte KG: & la parte <lb/>HD mouera&longs;&longs;i dalla leua CB, il cui &longs;o&longs;tegno è K, &longs;i che B con la leua CB cacci <lb/>la parte HD.<emph.end type="italics"/></s> </p> <figure id="fig111"></figure> <p type="main"> <s>Ma trouando&longs;i tre maniere di leue, come è &longs;tato di &longs;opra mo­<lb/>&longs;trato. </s> <s>però &longs;arà for&longs;e più conueneuole con&longs;iderare il cuneo <lb/>in que&longs;to modo. </s> </p> <p type="main"> <s><emph type="italics"/>Po&longs;te le co&longs;e iste&longs;&longs;e, intenda&longs;i la leua AB, & il &longs;o&longs;tegno &longs;uo B, & il pe&longs;o in H, come <lb/>nella &longs;econda di questo nella leua dicemmo. </s> <s>&longs;imilmente &longs;ia la leua CB co'l &longs;uo <lb/>&longs;o&longs;tegno B, & il pe&longs;o in K; &longs;iche la parte HD &longs;i moua dalla leua AB, il <lb/>cui &longs;o&longs;tegno è B, & il pe&longs;o in H; &longs;iche il punto H di e&longs;&longs;a leua AB cacci la <lb/>parte HD. & con modo &longs;imile la parte KG moua&longs;i dalla leua CB, il cut &longs;o&longs;te­<lb/>gno è B, & il pe&longs;o in K, &longs;iche il K di e&longs;&longs;a leua CB mouala parte KG. ilche <lb/>&longs;arà &longs;or&longs;e più conforme alla ragione.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Percioche &longs;ia il cuneo ABC; & &longs;iano due pe&longs;i &longs;eparati DEFG, & HIKL, fra <lb/>quali &longs;ia la parte DBH del cuneo, la cui cima B tenga il mezo tral'vno, & l'al­<lb/>tro &longs;ito. </s> <s>Percota&longs;i il cu <lb/>neo in modo, che anche <lb/>dauantaggio più &longs;ia cac <lb/>ciato fra i pe&longs;i, come pri <lb/>maè stato detto; per­<lb/>cioche &longs;ono que&longs;ti pe&longs;i <lb/>come &longs;e &longs;o&longs;&longs;ero vno con­<lb/>tinuo &longs;olamente GF <lb/>KL, che bi&longs;ogna&longs;&longs;e fen <lb/>dere: percioche nel mo­<lb/>do iste&longs;&longs;o la parte DG <lb/>mentre il cuneo è più ol <lb/>tre cacciato, &longs;i mouerà <lb/>ver&longs;o M, & la parte <lb/>HL ver&longs;o N. </s> <s>Moua&longs;i <lb/>dunque la parte DG <lb/>ver&longs;o M, & la parte <lb/>HL ver&longs;o N; & il B<emph.end type="italics"/><lb/> <arrow.to.target n="fig112"></arrow.to.target><lb/><emph type="italics"/>mentre trapa&longs;&longs;a più oltre, &longs;empre rimanga nel mezo tra l'vn pe&longs;o, & l'altro. </s> <s>Hor <lb/>mentre D G è mo&longs;&longs;o dal cuneo in uer&longs;o M; egli è manife&longs;to, che B non moue la <lb/>parte DG inuer&longs;o M con la leua CB, il cui &longs;o&longs;tegno è H, perche il punto B non <lb/>tocca il pe&longs;o; ma DG moueraßi dal punto D della leua con e&longs;&longs;a leua AB, che ha <lb/>per &longs;o&longs;tegno B; pereche il punto D tocca il pe&longs;o. </s> <s>& gli i&longs;trumenti mouono per <lb/>toccamento. </s> <s>&longs;imilmente HL moueraßi da H con la leua CB, che ha per &longs;o&longs;te­<lb/>gno B; & ambedue le leue &longs;i fanno re&longs;i&longs;tenza l'vna all'altra fra loro in B, talche <lb/>B faccia più tosto officio di &longs;o&longs;tegno, che di mouere il pe&longs;o. </s> <s>laqual co&longs;a anco ma­<lb/>nife&longs;teraßi in que&longs;ta maniera.<emph.end type="italics"/></s> </p> <pb pagenum="109"/> <figure id="fig112"></figure> <p type="main"> <s><emph type="italics"/>Sia quel che s'ha da fendere vnparallelo grammo rettangolo ABCD; & &longs;iano due <lb/>leue eguali EF GF, & le parti delle leue HF KF &longs;iano tra AB CD; & &longs;ia <lb/>HF eguale ad FK, & &longs;ia HA <lb/>eguale à KB. & faccia me&longs;tieri <lb/>con le leue EF FG fendere AB <lb/>CD &longs;enza perco&longs;&longs;a, cioè &longs;iano le <lb/>po&longs;&longs;anze mouenti in EG eguali. </s> <s><lb/>Ma per e&longs;&longs;ere fe&longs;&longs;a AB CD, egli <lb/>è me&longs;tieri che la parte HA &longs;i mo <lb/>ua ver&longs;o M; & KB ver&longs;o N: <lb/>ma mentre le leue &longs;i mouono, co­<lb/>me per e&longs;&longs;empio l'vna in M, & <lb/>l'altrain N; egli è nece&longs;&longs;ario, <lb/>che il punto F rimanga immobi <lb/>le, perche in e&longs;&longs;o &longs;i fa l'incontro del <lb/>le leue. </s> <s>Per laqual co&longs;a F &longs;arà il <lb/>&longs;o&longs;tegno dell'vna, & l'altra leua; <lb/>& FG mouerà la parte KB, il<emph.end type="italics"/><lb/> <arrow.to.target n="fig113"></arrow.to.target><lb/><emph type="italics"/>cui &longs;o&longs;tegno &longs;arà F, & la po&longs;&longs;anza mouente in G; & il pe&longs;o in K. &longs;imilmente <lb/>la parte HA moueraßi dalla leua EF, il cui &longs;o&longs;tegno è F, & la po&longs;&longs;anza in E, <lb/>& il pe&longs;o in H.<emph.end type="italics"/></s> </p> <figure id="fig113"></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e KH fo&longs;&longs;ero i &longs;o&longs;tegni immobili, & i pe&longs;i in F; mentre la leua FG &longs;i sforza di <lb/>mouere il pe&longs;o po&longs;to in F, all'hora le fa re&longs;i&longs;tenza la leua EF, laquale parimente <lb/>&longs;i sforza di mouere il pe&longs;o po&longs;to in F in uer&longs;o la parte oppo&longs;ta; ma percioche le po&longs;­<lb/>&longs;anze &longs;ono eguali, & le altre co&longs;e eguali: dunque non &longs;i farà mouimento in F; <lb/>percioche l'eguale non moue l'eguale. </s> <s>Egli è dunque pale&longs;e, che in F &longs;i fà grandißima <lb/>re&longs;i&longs;tenza dalle leue, che iui fra loro &longs;i incontrano, talche F viene ad e&longs;&longs;ere vn cer­<lb/>to che immobile. </s> <s>Per laqual co&longs;a con&longs;iderando il cuneo come moue con le leue fra <lb/>loro contrarie, egli per auentura le v&longs;a più to&longs;to à que&longs;to &longs;econdo modo, che al <lb/>primo.<emph.end type="italics"/></s> </p> <p type="main"> <s>Ma percioche tutto il cuneo &longs;i moue nel fendere, però po&longs;&longs;iamo <lb/>con&longs;iderarlo anche in vn'altro modo, cioè mentre che entra <lb/>in quel che viene fe&longs;&longs;o, niente altro e&longs;&longs;ere, che vn mouere vn <lb/>pe&longs;o &longs;opra vn piano inchinato all'orizonte. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia il piano egualmente di&longs;tante dall'orizonte, che paßi per AB; &longs;ia anco il cuneo <lb/>CDB; & &longs;ia CD eguale ad e&longs;&longs;a DB: & il lato del cuneo DB &longs;ia &longs;empre nel <lb/>&longs;ottopo&longs;to piano. </s> <s>&longs;ia dopo il pe&longs;o AEFG immobile in A; & &longs;ia la parte del <lb/>cuneo EDH &longs;otto AEFG. </s> <s>Hor percioche mentre il cuneo è perco&longs;&longs;o in CB, <lb/>maggior parte del detto cuneo entra &longs;otto AEFG, di quel che &longs;ia EDH; &longs;ia<emph.end type="italics"/><lb/> <arrow.to.target n="fig114"></arrow.to.target><lb/><emph type="italics"/>que&longs;ta parte IDH. & perche illato del cuneo DB è &longs;empre nel piano &longs;ottopo­<lb/>&longs;to tirato per AB egualmente di&longs;tante dall'orizonte, allhora quando la parte del <lb/>cuneo KDI &longs;arà &longs;otto AEFG; &longs;arà il punto K in H, & I &longs;otto E, ma IK <lb/>è maggiore di HE: dunque il punto E &longs;arà mo&longs;&longs;o in sù. </s> <s>& mentre il cuneo entra <lb/>&longs;otto AEFG, il punto E &longs;i mouerà in sù &longs;opra il lato EI del cuneo; & nel mo <lb/>do i&longs;te&longs;&longs;o, &longs;e il cuneo trapa&longs;&longs;erà più oltre, il punto E moueraßi &longs;empre &longs;opra il la­<lb/>to DC del cuneo; dunque il punto E del pe&longs;o &longs;i mouerà &longs;opra il piano DC in­<lb/>chinato all orizonte, la cui inclinatione è l'angolo BDC. che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig114"></figure> <p type="main"> <s><emph type="italics"/>In que&longs;to e&longs;&longs;empio con&longs;iderand<gap/> il cuneo, che moue à &longs;embianza dileua, egli è manife&longs;to <lb/>che il cuneo BCD moue il pe&longs;o AEFG conla leua CD: &longs;i che D &longs;ia il &longs;o&longs;te­<lb/>gno, & il pe&longs;o po&longs;to in E: ma non già con la leua BD, il cui &longs;o&longs;tegno &longs;ia H, & <lb/>il pe&longs;o po&longs;to in D.<emph.end type="italics"/></s> </p> <pb pagenum="110"/> <p type="main"> <s>Ma accioche la co&longs;a re&longs;ti più chiara v&longs;iamo altro e&longs;&longs;empio. </s> </p> <p type="main"> <s><emph type="italics"/>Sia vn piano egualmente di&longs;tante dall'orizonte, che paßi per AB: &longs;ia ilcuneo CAB, <lb/>il cui lato AB &longs;ia &longs;empre nel &longs;ottopo&longs;to piano; & &longs;ia il pe&longs;o AEFG, che non <lb/>habbia verun'altro moto &longs;e non in sù, & in giù ad ang oli retti all'orizonte: talcbe<emph.end type="italics"/><lb/> <arrow.to.target n="fig115"></arrow.to.target><lb/><emph type="italics"/>tirata la linea IGK à piombo del piano &longs;ottopo&longs;to, & di e&longs;&longs;a AB, il punto G <lb/>venga ad e&longs;&longs;ere &longs;empre nella linea IGK. </s> <s>& percio che mentre il cuneo è perco&longs;&longs;o <lb/>in CB, egli trapa&longs;&longs;a tutto più oltre &longs;opra AB; il pe&longs;o AEFG &longs;i leuerà, come <lb/>per le co&longs;e predette &longs;i è mo&longs;trato. </s> <s>Moua&longs;i il cuneo in modo, che E alla fine peruen <lb/>ga in C, & la giacitura del cuneo ABC venga ad e&longs;&longs;ere MNO, & la giaci­<lb/>tura del pe&longs;o AEFG &longs;ia PMQI, & G &longs;ia in I. </s> <s>co&longs;i perche mentre il cuneo <lb/>&longs;i moue &longs;opra la linea BO, il pe&longs;o AEFG &longs;i moue in sù dalla linea AC. & <lb/>mentre il cuneo ABC trapa&longs;&longs;a più oltre, il pe&longs;o AEFG &longs;empre più dal lato del <lb/>cuneo AC &longs;i leua: dunque il pe&longs;o AEFG &longs;i mouerà &longs;opra il piano del cuneo <lb/>AC; ilche veramente altro non è, &longs;e non vn piano inchinato all'orizonte, la cui <lb/>inclinatione è l'angolo BAC.<emph.end type="italics"/></s> </p> <figure id="fig115"></figure> <p type="main"> <s><emph type="italics"/>Que&longs;to mouimento &longs;i riduce ageuolmente alla bilancia, & alla leua; percioche quel <lb/>che &longs;i moue &longs;opra il piano inchinato all'orizonte, &longs;i riduce alla bilancia per la nona <lb/>propo&longs;itione di Pappo dell'ottauo libro delle raccolte matematiche. </s> <s>percioche è vna <lb/>i&longs;te&longs;&longs;a ragione, che ouero &longs;tando fermo il cuneo, il pe&longs;o &longs;i moua &longs;opra il lato del cu­<lb/>neo; ouero che e&longs;&longs;endo egli mo&longs;&longs;o, &longs;i moua anco il pe&longs;o &longs;opra il &longs;uo lato, come &longs;o­<lb/>pra vn piano inchinato all'orizonte.<emph.end type="italics"/></s> </p> <p type="main"> <s>La propo&longs;itione di Pappo allegata qui dall'Autore, & in altri luoghi di que&longs;to li­<lb/>bro, hò ripo&longs;ta in loco conueneuole nel Trattato della Vite, &longs;timando, che per <lb/>auentura ella &longs;ia per tornare più al propo&longs;ito della Vite, & &longs;eruirle in più chiarez­<lb/><gap/>a, che al Cuneo. </s> <s>Laquale pro po&longs;itione mi fù man data dall'Autore, & io &longs;e ben <lb/>non le manca nulla, la hò rincontrata accuratamente co'l Pappo Greco del Sig. </s> <s><lb/>Pinello, per modo che &longs;i haurà perfettis&longs;ima ad vtile, & diletto di coloro, i qua­<lb/>li niuna co&longs;a di Pappo &longs;crittore marauiglio&longs;o di Mecaniche hanno nè veduto, nè <lb/>letto giamai. </s> </p> <pb/> <p type="main"> <s>Hora mo&longs;triamo in che modo, quelle co&longs;e lequali &longs;ono fe&longs;&longs;e, &longs;i <lb/>mouano come &longs;opra piani inchinati all'orizonte. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il cuneo ABC, & AB &longs;ia eguale ad e&longs;&longs;a BC. </s> <s>Diuida&longs;i AC in due partii<gap/><lb/>D, & &longs;ia congiunta BD. &longs;ia dopo la linea EF, per laquale paßi il piano egual­<lb/>mente di&longs;tante dall'orizonte, & &longs;ia BD nella mede&longs;ima linea EF; & mentre il<emph.end type="italics"/><lb/> <arrow.to.target n="fig116"></arrow.to.target><lb/><emph type="italics"/>cuneo è perco&longs;&longs;o, & mentre &longs;i moue in ver&longs;o E, &longs;empre BD &longs;ia nella linea EF. <lb/>& quel che &longs;i hada fendere &longs;ia GHLM, dentro alquale &longs;ia la parte del cuneo <lb/>KBI: egli è mani&longs;e&longs;to, che mentre il cuneo &longs;i moue in ver&longs;o E, la parte KG mo­<lb/>uer&longs;i in ver&longs;o N; & la parte HI in ver&longs;o O. </s> <s>percota&longs;i il cuneo per modo che la <lb/>linea AC &longs;ia nella linea NO; allhora K &longs;arà in A, & I in C: & K perle <lb/>co&longs;e &longs;udette &longs;arà mo&longs;&longs;o &longs;opra KA, & I &longs;opra IC. </s> <s>Per laqual co&longs;a mentre &longs;i mo <lb/>ue il cuneo, la parte KG &longs;i mouerà &longs;opra il lato BA del cuneo, & la parte IH <lb/>&longs;opra il lato BC. </s> <s>La parte dunque KG &longs;i mouerà &longs;opra il piano inchinato all'o­<lb/>rizonte, la cui inclinatione è l'angolo FBA. &longs;imilmente IH &longs;i moue &longs;opra il <lb/>piano BC nell'angolo FBC. </s> <s>le parti dunque di quel che &longs;i &longs;ende moueran&longs;i &longs;o­<lb/>pra piani inchinati all'orizonte. </s> <s>& quantun que il piano BC &longs;ia &longs;otto l'orizonte; <lb/>tutta via la parte IH &longs;i moue &longs;opra IC, come &longs;e BC &longs;o&longs;&longs;e &longs;opra l'orizonte nel­<lb/>l'angolo DBC: percioche le parti di quel che &longs;i fende &longs;i mouono nel tempo me­<lb/>de&longs;imo dall'i&longs;te&longs;&longs;a po&longs;&longs;anza. </s> <s>&longs;arà dunque la mede&longs;ima ragione del mouimento della <lb/>parte IH, & della parte KF. </s> <s>&longs;imilmente è l'i&longs;te&longs;&longs;a ragione &longs;e EF è egualmente <lb/>di&longs;tante dall'orizonte, ouero &longs;e è à piombo dell'orizonte, ouero in altro modo: pe <lb/>roche egli è nece&longs;&longs;ario, che la po&longs;&longs;anza, laquale moue il cuneo, &longs;ia la mede&longs;ima, re­<lb/>&longs;tando le altre co&longs;e le mede&longs;ime. </s> <s>&longs;arà dunque la &longs;te&longs;&longs;a ragione.<emph.end type="italics"/></s> </p> <pb pagenum="111"/> <figure id="fig116"></figure> <p type="main"> <s><emph type="italics"/>Dopo que&longs;te co&longs;e egli è da con&longs;iderare, quali &longs;iano quelle co&longs;e, lequali fanno sì, che più <lb/>ageuolmente alcuna co&longs;a &longs;i moua, ouero &longs;i &longs;enda, lequali &longs;ono due.<emph.end type="italics"/></s> </p> <p type="main"> <s>Primieramente quel che opera in modo, che alcuna co&longs;a più <lb/>ageuolmente &longs;ia fe&longs;&longs;a. </s> <s>ilche più appartiene etiandio alla e&longs;­<lb/>&longs;enza del cuneo, è l'angolo po&longs;to alla cima del cuneo: pero che <lb/>quanto minore è l'angolo, tanto più ageuolmente moue, & <lb/>fende. </s> </p> <p type="main"> <s><emph type="italics"/>Siano due cunei ABC DEF, & l'angolo ABC po&longs;to alla cima &longs;ia minore dell'an <lb/>golo DEF. </s> <s>Dico che alcuna co&longs;a più ageuolmente &longs;i moue, ò fende dal cuneo <lb/>ABC, che da DEF. </s> <s>Diuidan&longs;i AC DF in due parti eguali ne'punti GH; <lb/>& &longs;iano congiunte BG & EH. </s> <s>Hor <lb/>percioche le parti di quello, che &longs;i fen <lb/>de dal cuneo ABC &longs;i mouono &longs;opra <lb/>il piano inchinato all'orizonte, la cui <lb/>inclinatione è GBA; & quelle che <lb/>dal cuneo DEF &longs;i mouono &longs;opra il <lb/>piano inchinato all'orizonte, la cui <lb/>inclinatione è HED, & l'angolo <lb/>GBA è minore dell'angolo HED; <lb/>per e&longs;&longs;ere GBA minore di DEF: <lb/>& per la nona di Pappo dell'ottauo <lb/>libro delle raccolte matematiche, quel <lb/>che &longs;i moue &longs;opra il piano AB, &longs;i mo <lb/>uerà più facilmente, & da po&longs;&longs;anza <lb/>minore, che &longs;opra ED. </s> <s>Quel che &longs;i <lb/>&longs;ende dunque dal cuneo ABC più<emph.end type="italics"/><lb/> <arrow.to.target n="fig117"></arrow.to.target><lb/><emph type="italics"/>ageuolmente, & da po&longs;&longs;anza minore &longs;i fende, che dal cuneo DEF. </s> <s>&longs;imilmente <lb/>mo&longs;trera&longs;&longs;i, che quanto più acuto &longs;arà l'angolo po&longs;to alla cima del cuneo, tanto più <lb/>ageuolmente mouera&longs;&longs;i, & fendera&longs;&longs;i alcuna co&longs;a. </s> <s>che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <figure id="fig117"></figure> <p type="main"> <s>Pos&longs;iamo dimo&longs;trare que&longs;to etiandio con altra ragione, con&longs;i­<lb/>derando il cuneo come egli moue con le leue contrarie l'vna <lb/>all'altra fra loro, fi come nel &longs;econdo modo fù detto. </s> <s>ma bi&longs;o <lb/>gna prima dimo&longs;trare que&longs;to. </s> </p> <p type="main"> <s><emph type="italics"/>Sia la leua AB, che habbia il &longs;uo &longs;o&longs;tegno B immobile, & quel che s'ha da mouere <lb/>&longs;ia CD EF rettangolo, co&longs;i di&longs;po&longs;to, che non po&longs;&longs;a mouer&longs;i in giù dalla parte di <lb/>FE; & il punto E &longs;ia immobile, & come centro; &longs;iche il punto D &longs;i moua per <lb/>la circonferenza del cerchio <lb/>DH, il cui centro &longs;ia E. & <lb/>C per la circonferenza CL, <lb/>&longs;i che la linea congiunta CE <lb/>&longs;ia il &longs;uo mezo diametro. </s> <s><lb/>di più CDEF tocchi la le <lb/>ua AB in C, & la leua <lb/>AB moua il pe&longs;o CDEF, <lb/>& la po&longs;&longs;anza mouente &longs;ia <lb/>in A, il &longs;o&longs;tegno in B, & <lb/>il pe&longs;o in C. &longs;ia dapoi vn'al <lb/>traleua MCN, laquale <lb/>etiandio moua CD EF, il <lb/>cui &longs;o&longs;tegno immobile &longs;ia <lb/>N; la po&longs;&longs;anza mouente in <lb/>M, & il pe&longs;o &longs;imilmente in <lb/>C; & &longs;ia CN eguale ad<emph.end type="italics"/><lb/> <arrow.to.target n="fig118"></arrow.to.target><lb/><emph type="italics"/>e&longs;&longs;a CB, & CM ad e&longs;&longs;a CA; & moua&longs;i alternamente il pe&longs;o CDEF con le <lb/>leue AB MN. </s> <s>Dico che CDEF più ageuolmente &longs;i mouerà dall'i&longs;te&longs;&longs;a po&longs;&longs;an <lb/>za con la leua AB, che con la leua MN.<emph.end type="italics"/></s> </p> <figure id="fig118"></figure> <p type="main"> <s><emph type="italics"/>Faccia&longs;i il centro B, & con lo &longs;patio BC de&longs;criua&longs;i la circonferenza CO. &longs;imilmen­<lb/>te co'l centro N, & lo &longs;patio NC de&longs;criua&longs;i la circonferenza CP. </s> <s>Hor percio­<lb/>che mentre la leua AB moue CD EF, il punto della leua C &longs;i moue &longs;opra la <lb/>circonferenza CO, per e&longs;&longs;ere B &longs;o&longs;tegno, & centro immobile. </s> <s>&longs;imilmente men­<lb/>tre la leua MN moue CD EF, il punto C &longs;i moue per la circonferenza CP: <lb/>mentre dunque la leua AB moue CD EF, &longs;i s&longs;orza mouere il punto C del pe&longs;o <lb/>&longs;opra la circonferenza CO; ilche non può già fare, perche C &longs;i moue &longs;opra la cir­<lb/>conferenza CL. </s> <s>Per laqual co&longs;a nel mouimento della leua AB &longs;econdo la parte<emph.end type="italics"/> <pb pagenum="112"/><emph type="italics"/>che le ri&longs;ponde, & nel mouimento del pe&longs;o fatto &longs;econdo C, ne na&longs;ce vn certo con­<lb/>tra&longs;to: percioche &longs;i mouono in diuer&longs;e parti. </s> <s>&longs;imilmente mentre la leua MN mo <lb/>ue CD EF, &longs;i sforza mouere il C &longs;opra la circonferenza CP: & però in que&longs;to <lb/>ancora na&longs;ce in ambidue i mouimenti vn &longs;imile contra&longs;to. </s> <s>Et perche la circon&longs;eren <lb/>za CO è più da pre&longs;&longs;o alla circonferenza CL, che non è CP, cioè più da pre&longs;&longs;o <lb/>al mouimento, che fa il punto C del pe&longs;o; però il contra&longs;to tra il mouimento della le <lb/>ua AB, & il mouimento del pe&longs;o C &longs;arà minore, che tra il mouimento della leua <lb/>MN, & il mouimento dell'i&longs;te&longs;&longs;o C, ilche etiandio è chiaro, &longs;e &longs;i intenda che CF <lb/>&longs;ia à piombo dell'orizonte; percioche all'hora la circonferenza CP più inchina al <lb/>ba&longs;&longs;o, che CO: & CL và in sù. </s> <s>& perciò &longs;i fa contra&longs;to minore tra la leua <lb/>AB, & il mouimento C, che fra la leua MN, & il mouimento C. </s> <s>Ma doue <lb/>è conte&longs;a minore, iui è più ageuolezza. </s> <s>Dunque &longs;i mouerà più facilmente CDEF <lb/>con la leua AB, che con la leua MN. che bi&longs;ognaua mo&longs;trare.<emph.end type="italics"/></s> </p> <p type="head"> <s>COROLLARIO</s> </p> <p type="main"> <s>Da que&longs;to è chiaro, che quanto minore è l'angolo contenuto <lb/>dalla linea CF, ouero CE, ouero CD; cioè quanto <lb/>minore è l'angolo BCF, ouero BCE, ouero anche BCD; <lb/>tanto più ageuolmente il pe&longs;o è mo&longs;&longs;o. </s> <s>ilche mo&longs;treras&longs;i nel­<lb/>l'i&longs;te&longs;&longs;o modo. </s> </p> <p type="main"> <s><emph type="italics"/>Siano li cunei ABC DEF, & l'angolo ABC &longs;ia minore dell'angolo DEF, & <lb/>AB BC DE EF &longs;iano tra loro eguali. </s> <s>&longs;iano dapoi quattro pe&longs;i eguali GH IL <lb/>NO QR rettangoli; & &longs;iano LM KH nella mede&longs;ima linea retta. </s> <s>&longs;imilmente<emph.end type="italics"/><lb/> <arrow.to.target n="note309"></arrow.to.target> <emph type="italics"/>RS PO in linea retta; &longs;aranno GK IM egualmente di&longs;tanti, & NP QS an <lb/>co egualmente di&longs;tanti. </s> <s>&longs;ia IBG la parte del cuneo fraipe&longs;i GH IL; & la par <lb/>te del cuneo QEN fra i pe&longs;i NOQR; & &longs;iano IB BG QE EN traloro <lb/>eguali. </s> <s>Dico che i pe&longs;i GH IL più ageuolmente &longs;aranno dalla po&longs;&longs;anza i&longs;te&longs;&longs;a co'l <lb/>cuneo ABC mo&longs;si, che i pe&longs;i NO QR dal cuneo DEF.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note309"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 28. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="main"> <s><emph type="italics"/>Diuidan&longs;i AC DF in due parti eguali in TV, & congiungan&longs;i TBVE, &longs;aranno <lb/>gli angoli po&longs;ti al T, & V retti. </s> <s>congiunga&longs;i IG, laquale tagli BT in X. </s> <s>Hor<emph.end type="italics"/><lb/> <arrow.to.target n="fig119"></arrow.to.target><lb/> <arrow.to.target n="note310"></arrow.to.target> <emph type="italics"/>percioche IB è eguale à BG, & BA eguale à BC: &longs;arà IA eguale ad e&longs;&longs;a<emph.end type="italics"/><lb/> <arrow.to.target n="note311"></arrow.to.target> <emph type="italics"/>GC. </s> <s>Per laqual co&longs;a BI ad IA è co&longs;i, come BG à GC; dunque IG è egualmen <lb/>te di&longs;tante ad e&longs;&longs;a AC: & perciò gli angoli ad X &longs;ono retti; ma gli angoli XGK<emph.end type="italics"/><lb/> <arrow.to.target n="note312"></arrow.to.target> <emph type="italics"/>XIM &longs;ono retti, peroche GM è rettangolo. </s> <s>Per laqual co&longs;a TB è egualmente di­<lb/>&longs;tante da GKIM. </s> <s>dunque l'angolo TBC è eguale all'angolo BGK, & TBA è <lb/>eguale ad e&longs;&longs;o BIM. <expan abbr="&longs;imilmēte">&longs;imilmente</expan> mo&longs;treremo che l'angolo VEF è eguale ad ENP, <lb/>& VED eguale ad EQS. & per e&longs;&longs;ere l'angolo ABC minore dell'angolo DEF; <lb/>&longs;arà anco l'angolo TBC minore di VEN. </s> <s>Per laqual co&longs;a BGK &longs;arà anche mi­<lb/>nore di ENP. </s> <s>con &longs;imile modo BIM è minore di EQS. </s> <s>Hor percioche il cuneo <lb/>ABC moue con due leue AB BC, che hanno i &longs;o&longs;tegni &longs;uoi in B, & i pe&longs;i in <lb/>GI. &longs;imilmente il cuneo DEF moue con due altre leue DE EF, i cui &longs;o&longs;tegni &longs;o­<lb/>no in E; & i pe&longs;i in NQ: per la precedente i pe&longs;i GH IL &longs;i moueranno più ageuol <lb/>mente con le leue AB BC, che i pe&longs;i NO QR conle leue DE EF. </s> <s>i pe&longs;i dunque <lb/>GH IL, &longs;i moueranno più ageuolmente co'lcuneo ABC, che i pe&longs;i NO QR co'l<emph.end type="italics"/> <pb pagenum="113"/><emph type="italics"/>euneo DEF. & perche è la ragione i&longs;le&longs;&longs;a nel mouere & nel fendere; però più age <lb/>uolmente &longs;i fenderà alcuna co&longs;a co'l cuneo ABC, che co'l cuneo DE<emph.end type="italics"/>F. <emph type="italics"/>Et dimo­<lb/>&longs;trera&longs;&longs;i mede&longs;imamente che quanto minore è l'angolo po&longs;to alla cima del cuneo, tan <lb/>to più ageuolmente &longs;i moue alcuna co&longs;a, ouero &longs;i fende, che bi&longs;ognaua mostrare.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note310"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 2. <emph type="italics"/>del &longs;esto.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note311"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 9. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <p type="margin"> <s><margin.target id="note312"></margin.target><emph type="italics"/>Per la<emph.end type="italics"/> 28. <emph type="italics"/>del primo.<emph.end type="italics"/></s> </p> <figure id="fig119"></figure> <p type="main"> <s><emph type="italics"/>Oltre à ciò quelle co&longs;e, lequali &longs;ono mo&longs;&longs;e dal cuneo DEF, &longs;i mouono per maggiori <lb/>&longs;patij che quelle che &longs;ono mo&longs;&longs;e dal cuneo ABC. </s> <s>Imperoche affine che D<emph.end type="italics"/>F <emph type="italics"/>&longs;ia <lb/>tra QN, & affine che AC &longs;ia trà IG, egli è nece&longs;&longs;ario che QN &longs;i mouano per <lb/>maggiori &longs;patij, cioè l'vno alla de&longs;tra, l'alrro alla &longs;ini&longs;tra, che IG, per e&longs;&longs;ere D<emph.end type="italics"/>F <lb/><emph type="italics"/>maggiore di AC: pur che tutto il cuneo entri fra i pe&longs;i. </s> <s>Ma dalla po&longs;&longs;anza più <lb/>facilmente &longs;i moue per minor &longs;patio alcuna co&longs;a nel mede&longs;imo tempo, che per mag­<lb/>giore: pur che le altre co&longs;e con le quali &longs;i fà il mouimento &longs;iano eguali: &longs;e dunque <lb/>AC D<emph.end type="italics"/>F <emph type="italics"/>peruerranno nell'i&longs;te&longs;&longs;o tempo in IG QN, e&longs;&longs;endo A I CG DQ<emph.end type="italics"/>F<emph type="italics"/>N <lb/>tra loro eguali; più facilmente dalla po&longs;&longs;anza &longs;imoueranno GI co'l cuneo ABC, <lb/>che QN co'l cuneo DE<emph.end type="italics"/>F. <emph type="italics"/>per laqual co&longs;a i pe&longs;i GHIL &longs;i moueranno più facil­<lb/>mente dalla po&longs;&longs;anza co'l cuneo ABC, che i pe&longs;i NO QR co'l cuneo DE<emph.end type="italics"/>F. <lb/><emph type="italics"/>& &longs;imilmente &longs;i mostrerà, che quanto l'angolo po&longs;to alla cima del cuneo &longs;arà mino <lb/>re, tanto più ageuolmente &longs;i moueranno i pe&longs;i, ouero &longs;i fenderanno.<emph.end type="italics"/></s> </p> <p type="main"> <s>La &longs;econda co&longs;a laquale è cagione, che alcuna co&longs;a &longs;i fenda più <lb/>ageuolmente è la perco&longs;&longs;a, mediante laquale è mo&longs;&longs;o il cuneo <lb/>& moue, cioè vien perco&longs;&longs;o, & fende. </s> </p> <pb/> <p type="main"> <s><emph type="italics"/>Sia il cuneo A, quel <lb/>che s'hada fendere <lb/>B, & quel che per <lb/>cuote C; ilquale <lb/>ouero da &longs;e &longs;te&longs;&longs;o <lb/>percuote, & moue; <lb/>ouero dalla po&longs;&longs;an­<lb/>za che lo regge, & <lb/>moue. </s> <s>che &longs;e da &longs;e <lb/>&longs;te&longs;&longs;o, prima s ha da <lb/>auertire, che quan­<lb/>to più &longs;arà graue, <lb/>tanto &longs;i farà la per <lb/>co&longs;&longs;a maggiore. </s> <s>& <lb/>oltre à ciò quanto <lb/>più &longs;arà lunga la di­<lb/>stanzatra AC, fa <lb/>raßi parimente mag <lb/>giore perco&longs;&longs;a: pe­<lb/>roche cia&longs;cuna co&longs;a <lb/>graue, mentre &longs;i mo <lb/>ue, prende più di gra <lb/>uezza mo&longs;&longs;a, che <lb/>&longs;tando ferma, & <lb/>dauantaggio anco <lb/>più, quanto più da <lb/>lontano è mo&longs;&longs;a.<emph.end type="italics"/></s> </p> <figure></figure> <p type="main"> <s><emph type="italics"/>Che &longs;e C &longs;arà mo&longs;&longs;o da qualche po&longs;&longs;anza, <lb/>come per lo manico DE &longs;ia mo&longs;&longs;o. </s> <s>Pri <lb/>ma quanto C &longs;arà più graue; dapoi <lb/>quanto &longs;arà più lungo DE, tanto la <lb/>perco&longs;&longs;a faraßimaggiore: percioche &longs;e <lb/>la po&longs;&longs;anza mouente &longs;arà posta in E, <lb/>&longs;arà il C più di&longs;tante dal centro, & pe <lb/>rò moueraßi più tosto, come Ari&longs;to­<lb/>tele dimostra nelle questioni mecani­<lb/>che; & puote e&longs;&longs;ere anco chiaro da <lb/>quelle co&longs;e, che furono dette nel trat­<lb/>tato della bilancia, che quanto più il<emph.end type="italics"/><lb/> <arrow.to.target n="fig120"></arrow.to.target> <pb pagenum="114"/><emph type="italics"/>pe&longs;o C è di&longs;tante dal centro, tanto più far&longs;i graue, & vrterà etiandio con più ga­<lb/>gliard'empito, e&longs;&longs;endo la forza in E più po&longs;&longs;ente.<emph.end type="italics"/></s> </p> <figure id="fig120"></figure> <p type="main"> <s><emph type="italics"/>Ma que&longs;ta è la <expan abbr="&longs;ecõda">&longs;econda</expan> co&longs;a, laqual è cagione che con que&longs;to i&longs;trumento &longs;imouano gran <lb/>pe&longs;i, & &longs;i fendano. </s> <s>Percioche la perco&longs;&longs;a è vna forza gagliardißima, come è ma­<lb/>nife&longs;to da la decimanona delle questioni <lb/>mec aniche di Ari&longs;totele: peroche &longs;e &longs;o­<lb/>pra il cuneo &longs;i imporrà vn pe&longs;o grandißi­<lb/>mo, allhora il cuneo non farà nulla à pa­<lb/>ragone &longs;petialmente della perco&longs;&longs;a. </s> <s>che &longs;e <lb/>anco &longs;i adatta&longs;&longs;e al cuneo vna leua, ouero <lb/>vna vite, ò qualche altro tale &longs;tromento <lb/>per cacciare il cuneo più à dentro nel pe&longs;o, <lb/>non auenir à effetto qua&longs;i di momento niu <lb/>no, ri&longs;petto alla perco&longs;&longs;a. </s> <s>della qual co&longs;a <lb/>puote e&longs;&longs;ere inditio, che &longs;e fo&longs;&longs;e il corpo A <lb/>di pietra, da cui alcuno vole&longs;&longs;e leuar via<emph.end type="italics"/><lb/> <arrow.to.target n="fig121"></arrow.to.target><lb/><emph type="italics"/>qualche parte, come vn pezzo dell'angolo B, allhora potrebbe rompere ageuolmen <lb/>te con vno martello di ferro, &longs;enza altro &longs;tromento, percotendo in B, qualche pezzo <lb/>dell'angolo B: ilche non potrà fare con ne&longs;&longs;uno altro &longs;tromento, che &longs;ia priuo di per­<lb/>co&longs;&longs;a, &longs;e non con difficultà grandißima, &longs;ia ò leua, ò vite, ò qual &longs;i voglia altra co&longs;a <lb/>tale. </s> <s>La onde la perco&longs;&longs;a è cagione, che &longs;i fendano i gran pe&longs;i. </s> <s>& hauendo la per­<lb/>co&longs;&longs;a co&longs;i gran forza, &longs;e le ag giungeremo qualche &longs;tromento accommodato à moue­<lb/>re, & fendere, vedremo per certo co&longs;e marauiglio&longs;e. </s> <s>Cote&longs;to &longs;tromento è il cuneo, <lb/>nel quale due co&longs;e, inquanto s'ap <lb/>partiene alla &longs;ua forma, occor­<lb/>rono ad e&longs;&longs;ere con&longs;iderate: L'v­<lb/>na, che il cuneo è attißimo à ri­<lb/>ceuere, & &longs;o&longs;tenere la perco&longs;&longs;a: <lb/>l'altra è, che per la &longs;ua &longs;ottigliez <lb/>za nell vna delle parti <expan abbr="facilmēte">facilmente</expan> <lb/>entra ne'corpi, come e&longs;pre&longs;&longs;a­<lb/>mente &longs;i vede. </s> <s>Il cuneo dunque <lb/>opera&longs;i con la &longs;ua perco&longs;&longs;a, che <lb/>vediamo qua&longs;i miracoli nel fen­<lb/>dere i corpi.<emph.end type="italics"/></s> </p> <figure id="fig121"></figure> <figure></figure> <pb/> <p type="main"> <s><emph type="italics"/>Alla facoltà di cotale &longs;tromento &longs;i po&longs;&longs;ono etiandio ridurre commodamente quelle co <lb/>&longs;e tutte, lequali con perco&longs;&longs;a, ouero &longs;pinta tagliano, diuidono, &longs;orano, & fanno al <lb/>tri cotali e&longs;fetti, come &longs;pade, punte, coltelli, &longs;curi, & &longs;imili. </s> <s>La &longs;ega ancora &longs;i <lb/>ridurrà à que&longs;to: peroche i &longs;uoi denti percotono, & &longs;ono à &longs;emb ianza di cuneo.<emph.end type="italics"/></s> </p> <p type="head"> <s>IL FINE DEL CVNEO.</s> </p> <pb pagenum="115"/> <p type="head"> <s>DELLA VITE.</s> </p> <figure></figure> <p type="main"> <s>Pappo nell'i&longs;te&longs;&longs;o ottauo libro trattando mol­<lb/>te co&longs;e della vite, in&longs;egna come ella &longs;i deue fa­<lb/>bricare; & come con cotale &longs;tromento &longs;i moua­<lb/>no grandi pe&longs;i: & di più mette altre &longs;peculatio­<lb/>ni molto vtili alla cognitione di lei. </s> <s>Ma per­<lb/>cioche tra le altre co&longs;e egli promette di voler mo&longs;trare la vi­<lb/>te niente altro e&longs;&longs;ere, che vn cuneo pre&longs;o &longs;enza la perco&longs;&longs;a, il <lb/>quale faccia il mouimento &longs;uo con la leua. </s> <s>& que&longs;to in lui &longs;i <lb/>de&longs;idera: però noi &longs;i sforzeremo di mo&longs;trare ciò. </s> <s>& di più <lb/>ridurre la detta vite alla leua, & alla bilancia, accioche alla fi­<lb/>ne &longs;e n'habbia compiuta cognitione. </s> </p> <p type="main"> <s>Hò ritonuto nel tradurre le parole Cilindro, & Helice i vocaboli i&longs;tes&longs;i, come l'Au­<lb/>tore gli ha po&longs;ti, percioche la no&longs;tra lingua pouera ancora di que&longs;te voci, non ne <lb/>hà fin hora approuata alcuna per buona, & communemente in te&longs;a in tutta Italia <lb/>per &longs;ignificare le predette due co&longs;e Cilindro, & Helice. </s> <s>Però io, affine di dome­<lb/>&longs;ticarle, hò voluto farne e&longs;perientia, la&longs;ciandole co&longs;i, &longs;e per auentura pote&longs;&longs;ero <lb/>e&longs;&longs;er accettate. </s> <s>Cilindro, voce Greca, è quel ba&longs;tone lauorato al torno, nel quale <lb/>&longs;i intagliano quei rileui co' &longs;uoi concaui, che vanno montando in &longs;u&longs;o à lumaca, <lb/>ò chio cciola, & &longs;i dicono vite, ouero in qualche contrada d'Italia vermi, ò chioc <lb/>ciole, & l'Autore qui noma Hlici. </s> <s>Ba&longs;ta che la co&longs;a re&longs;ti chiara, non que&longs;tionan­<lb/>do de' nomi, & &longs;i intenda che voglia dire Cilindro, & Helice. </s> <s>La Vite in latino <lb/>&longs;i chiama Cochlea à &longs;imiglianza cied'io dell'animale che &longs;i <expan abbr="mãgia">mangia</expan> detto lumaca, ò <lb/>bouolo, ò chiocciola, che è più &longs;imile à Cochlea latino, talche la vite, &longs;tando sù <lb/>inomi, viene ad hauere pre&longs;o il nome da quell'animale, che nella ca&longs;a, la quale &longs;em <lb/>pre porta &longs;eco &longs;i ra&longs;&longs;embra, mas&longs;imamente nel fondo di e&longs;&longs;a, in certo modo al rile <lb/>uo, ò verme, ouero helice della vite. </s> <s>Onde ben &longs;i potrebbe con ragione dire <lb/>chiocciola alla vite, volgarizando il vocabolo latino cochlea, come &longs;i appellane <lb/>chiocciole le &longs;cale che a&longs;cendono à vite. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il cuneo ABC, ilquale &longs;iriuolga d'intorno al Cilindro DE, & &longs;ia IGH il cu­<lb/>neo riuolto d'intorno al cilindro, la cui cima &longs;ia I. &longs;ia dapoi il cilindro in&longs;ieme co'lcu <lb/>neo po&longs;toui d'intorno accomm o dato in modo, che &longs;enza alcuno impedimento &longs;ipo&longs;&longs;a <lb/>volgere intorno co'l manico KF attaccato all'a&longs;&longs;e: & &longs;ia LMNO quel che s'ha <lb/>da fendere, ilquale etiandio dalla parte di MN &longs;ia immobile, &longs;i come &longs;uole far&longs;i <lb/>in quelle co&longs;e, che &longs;i fendono. </s> <s>& &longs;ia la cima I'tra RS. </s> <s>Volga&longs;i intorno KF, &<emph.end type="italics"/><lb/> <arrow.to.target n="fig122"></arrow.to.target><lb/><emph type="italics"/>peruenga à KP; & mentre che KF &longs;i volge intorno, tutto il cilindro DE anc<gap/><lb/>ra &longs;i volge intorno, & il cuneo IGH. per laqual co&longs;a mentre KF &longs;arà in KP, <lb/>la cima I non &longs;arà più tra RS, ma altr a parte del cuneo, come TV: ma TV è <lb/>maggiore di RS; peroche la parte del cuneo, laquale è più di&longs;tante dalla cima, &longs;em­<lb/>pre è mag giore di quella, che è più ad e&longs;&longs;a vicina. </s> <s>accioche dunque TV &longs;ia tra RS, <lb/>bi&longs;ogna che R ceda, & &longs;i moua ver&longs;o X, & S in ver&longs;o Z, come fanno le co&longs;e, che <lb/>&longs;i fendono. </s> <s>tutto dunque LMNO &longs;i fenderà. </s> <s>Similmente dimo&longs;treremo, che men <lb/>tre il manico KP &longs;arà in KQ, allhora GH &longs;arà fra RS: & mentre GH &longs;arà <lb/>tra RS, egli è nece&longs;&longs;ario che R &longs;ia in X, & S in Z. talche XZ &longs;ia eguale à GH; <lb/>& &longs;empre LM NO &longs;i fenderà dauantaggio. </s> <s>co&longs;i dunque è manife&longs;to, che mentre <lb/>KF &longs;i volge intorno, &longs;empre R &longs;i moue in ver&longs;o X, & S in ver&longs;o Z: & R mo­<lb/><gap/>er&longs;i &longs;empre &longs;opra ITG, & S &longs;opra IVH, cioè &longs;opra i lati del cuneo volti <lb/>d'intorno al cilindro.<emph.end type="italics"/></s> </p> <pb pagenum="116"/> <figure id="fig122"></figure> <p type="head"> <s>PROPOSITIONE I.</s> </p> <p type="main"> <s>Il cuneo accommodato in que&longs;to modo d'intorno al cilindro, <lb/>niente altro è, che la vite, laquale habbia due helici congiun <lb/>te fra loro in vno punto. </s> </p> <p type="main"> <s><emph type="italics"/>Sia il cuneo ABC; & AB &longs;ia eguale à BC. </s> <s>diuida&longs;i AC in due parti in D, <lb/>& congiunga&longs;i BD; &longs;arà BD à piombo di AC: & AD eguale à DC, & il <lb/>triangolo ABD eguale al triangolo CBD. </s> <s>Faccia&longs;i dapoi i triangoli rettangoli <lb/>EFG HIK non &longs;olo tra loro eguali, ma etiandio eguali ad ambidue i triang oli<emph.end type="italics"/><lb/> <arrow.to.target n="fig123"></arrow.to.target><lb/><emph type="italics"/>ADB, & CDB. & &longs;ia il cilindro LMNO, la cui linea che lo circonda detto <lb/>Perimetro &longs;ia eguale ad ambedue FGKI: & LMNO &longs;ia parallelo gram­<lb/>mo per l'a&longs;&longs;e. </s> <s>& faccia&longs;i MP eguale ad FE, & PN eguale ad HI. & pon <lb/>ga&longs;i HI in NP, & inuolga&longs;i il triangolo HIK d'intorno al cilindro; & &longs;ia de <lb/>&longs;critta la helice NQR &longs;econdo KH, come in&longs;egna anche Pappo nell'ottauo libro <lb/>alla propo&longs;itione vige&longs;ima quarta. </s> <s>& &longs;imilmente ponga&longs;i EF in MP, & in­<lb/>uolga&longs;i il triangolo EFG d'intorno al cilindro, & de&longs;criua&longs;i per EG la helice <lb/>PRM. & co&longs;i per e&longs;&longs;ere PM PN eguali ad EF HI, &longs;arà MN eguale ad <lb/>e&longs;&longs;a AC, & per e&longs;&longs;ere le helici PRM PQN eguali alle linee EG HK; &longs;a-<emph.end type="italics"/></s> </p> <figure id="fig123"></figure> <p type="main"> <s><emph type="italics"/>Similm ente &longs;e la vite <lb/>haurà più helici co­<lb/>me nella &longs;econda &longs;i­<lb/>gura, il pe&longs;o A, men <lb/>tre la vite &longs;i volge <lb/>intorno, &longs;empre &longs;i <lb/>mouerà &longs;opr a le he­<lb/>lici BCD EFG; <lb/>pur che il pe&longs;o A <lb/>in modo &longs;i adatti, <lb/>che non po&longs;&longs;a mo­<lb/>uer&longs;i &longs;e non &longs;opra la <lb/>retta linea HI e­<lb/>gualmente di&longs;tante <lb/>da e&longs;&longs;o cilindro. </s> <s>Per <lb/>cioche nell'i&longs;te&longs;&longs;o mo <lb/>do, che &longs;i moue &longs;o­<lb/>pra la prima helice, <lb/>&longs;i moue etiandio &longs;o­<lb/>pra la &longs;econda, & &longs;o <lb/>pra la terza, et &longs;opra <lb/>le altre. </s> <s>Percioche <lb/>quante &longs;i <expan abbr="vogliāheli">voglianheli</expan> <lb/>ci che &longs;iano, non &longs;on <lb/>altro niente, che vn <lb/>lato del cuneo inuol­<lb/>to d'intorno all'i&longs;te&longs;-<emph.end type="italics"/><lb/> <arrow.to.target n="fig124"></arrow.to.target><lb/><emph type="italics"/>&longs;o cilindro vna, & più volte. </s> <s>et &longs;ia la vite ouero à piombo dell'orizonte, ouero egual <lb/>mente di&longs;tante dall orizonte, ouero in altro modo collocata, nonimporta nulla; per­<lb/>cioche &longs;empre valerà l'i&longs;te&longs;&longs;a ragione.<emph.end type="italics"/></s> </p> <pb pagenum="117"/> <figure id="fig124"></figure> <p type="main"> <s><emph type="italics"/>che non po&longs;&longs;ano mouer&longs;i &longs;e non &longs;opra la diritta linea LO, laquale &longs;ia egualmente <lb/>di&longs;tante dail'a&longs;&longs;e del cilindro; & &longs;iano MN pre&longs;&longs;o la cima I del cuneo. </s> <s>Volga&longs;i <lb/>intorno KF, & peruenga in KP: & mentre KF &longs;arà in KP, allhora TV &longs;a <lb/>rà fra i pe&longs;i MN, &longs;i come di &longs;opra habbiamo detto. </s> <s>dunque M &longs;i mouerà ver&longs;o<emph.end type="italics"/><lb/> <arrow.to.target n="fig125"></arrow.to.target><lb/><emph type="italics"/>L, & N ver&longs;o O. </s> <s>Similmente mo&longs;trera&longs;&longs;i, che mentre KP &longs;arà in KQ, allho­<lb/>ra GH &longs;arà tra i pe&longs;i MN; & M &longs;arà in X, & N in Z; &longs;i che XZ &longs;arà <lb/>eguale <gap/> GH. </s> <s>Per laqual co&longs;a mentre KF &longs;i volge intorno, &longs;empre il pe&longs;o N &longs;i <lb/>moue in ver&longs;o O, & &longs;opra la helice IRS; & M &longs;opra l'altra helice.<emph.end type="italics"/></s> </p> <pb/> <figure id="fig125"></figure> <p type="main"> <s><emph type="italics"/>ranno dunque le dette helici eguali ad e&longs;&longs;e AB BC. dunque il cuneo. </s> <s>ABC &longs;arà <lb/>tutto inuolt<foreign lang="greek">q</foreign> d'intorno al cilindro LMNO. </s> <s>Siano tagliate da poile helici, come <lb/>in&longs;egna Pappo, &longs;econdo la larghezza del cuneo; & à que&longs;lo modo il cuneo in&longs;ieme<emph.end type="italics"/><lb/> <arrow.to.target n="fig126"></arrow.to.target><lb/><emph type="italics"/>co'l cilindro niente altro &longs;arà, che la vite, laqual